{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0001","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"有序四則運算結果","stem_zh":"有理數Q；a=1/2、b=0/1。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。","answer":{"kind":"expression","value":"(1/2, 1/2, 0/1, 未定義)","explanation_zh":"有序四則結果=(1/2, 1/2, 0/1, 未定義)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有序四則運算結果"]},"variant":{"family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112001,"parameters":{"a":[1,2],"b":[0,1],"operation":"density11_arithmetic_ordered_four_results","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"1/2\",\"0/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0002","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"有序四則運算結果","stem_zh":"有理數Q；a=0/1、b=-2/3。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。","answer":{"kind":"expression","value":"(-2/3, 2/3, 0/1, 0/1)","explanation_zh":"有序四則結果=(-2/3, 2/3, 0/1, 0/1)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有序四則運算結果"]},"variant":{"family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112002,"parameters":{"a":[0,1],"b":[-2,3],"operation":"density11_arithmetic_ordered_four_results","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"0/1\",\"-2/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0003","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"有序四則運算結果","stem_zh":"有理數Q；a=-10/3、b=3/4。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。","answer":{"kind":"expression","value":"(-31/12, -49/12, -5/2, -40/9)","explanation_zh":"有序四則結果=(-31/12, -49/12, -5/2, -40/9)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有序四則運算結果"]},"variant":{"family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112003,"parameters":{"a":[-10,3],"b":[3,4],"operation":"density11_arithmetic_ordered_four_results","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"-10/3\",\"3/4\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0004","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"有序四則運算結果","stem_zh":"有理數Q；a=-9/1、b=-4/5。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。","answer":{"kind":"expression","value":"(-49/5, -41/5, 36/5, 45/4)","explanation_zh":"有序四則結果=(-49/5, -41/5, 36/5, 45/4)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有序四則運算結果"]},"variant":{"family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112004,"parameters":{"a":[-9,1],"b":[-4,5],"operation":"density11_arithmetic_ordered_four_results","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"-9/1\",\"-4/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0005","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"有序四則運算結果","stem_zh":"有理數Q；a=-4/1、b=5/2。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。","answer":{"kind":"expression","value":"(-3/2, -13/2, -10/1, -8/5)","explanation_zh":"有序四則結果=(-3/2, -13/2, -10/1, -8/5)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有序四則運算結果"]},"variant":{"family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112005,"parameters":{"a":[-8,2],"b":[5,2],"operation":"density11_arithmetic_ordered_four_results","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"-4/1\",\"5/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0006","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"有序四則運算結果","stem_zh":"有理數Q；a=-7/3、b=-2/1。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。","answer":{"kind":"expression","value":"(-13/3, -1/3, 14/3, 7/6)","explanation_zh":"有序四則結果=(-13/3, -1/3, 14/3, 7/6)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有序四則運算結果"]},"variant":{"family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112006,"parameters":{"a":[-7,3],"b":[-6,3],"operation":"density11_arithmetic_ordered_four_results","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"-7/3\",\"-2/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0007","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"有序四則運算結果","stem_zh":"有理數Q；a=-6/1、b=7/4。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。","answer":{"kind":"expression","value":"(-17/4, -31/4, -21/2, -24/7)","explanation_zh":"有序四則結果=(-17/4, -31/4, -21/2, -24/7)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有序四則運算結果"]},"variant":{"family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112007,"parameters":{"a":[-6,1],"b":[7,4],"operation":"density11_arithmetic_ordered_four_results","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"-6/1\",\"7/4\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0008","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"有序四則運算結果","stem_zh":"有理數Q；a=-5/2、b=-8/5。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。","answer":{"kind":"expression","value":"(-41/10, -9/10, 4/1, 25/16)","explanation_zh":"有序四則結果=(-41/10, -9/10, 4/1, 25/16)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["有序四則運算結果"]},"variant":{"family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112008,"parameters":{"a":[-5,2],"b":[-8,5],"operation":"density11_arithmetic_ordered_four_results","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"-5/2\",\"-8/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0009","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"有序四則運算結果","stem_zh":"有理數Q；a=-4/3、b=9/2。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。","answer":{"kind":"expression","value":"(19/6, -35/6, -6/1, -8/27)","explanation_zh":"有序四則結果=(19/6, -35/6, -6/1, -8/27)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有序四則運算結果"]},"variant":{"family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112009,"parameters":{"a":[-4,3],"b":[9,2],"operation":"density11_arithmetic_ordered_four_results","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"-4/3\",\"9/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0010","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"有序四則運算結果","stem_zh":"有理數Q；a=-3/1、b=-10/3。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。","answer":{"kind":"expression","value":"(-19/3, 1/3, 10/1, 9/10)","explanation_zh":"有序四則結果=(-19/3, 1/3, 10/1, 9/10)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有序四則運算結果"]},"variant":{"family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112010,"parameters":{"a":[-3,1],"b":[-10,3],"operation":"density11_arithmetic_ordered_four_results","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"-3/1\",\"-10/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0011","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"有序四則運算結果","stem_zh":"有理數Q；a=-1/1、b=11/4。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。","answer":{"kind":"expression","value":"(7/4, -15/4, -11/4, -4/11)","explanation_zh":"有序四則結果=(7/4, -15/4, -11/4, -4/11)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有序四則運算結果"]},"variant":{"family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112011,"parameters":{"a":[-2,2],"b":[11,4],"operation":"density11_arithmetic_ordered_four_results","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"-1/1\",\"11/4\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0012","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"有序四則運算結果","stem_zh":"有理數Q；a=-1/3、b=-12/5。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。","answer":{"kind":"expression","value":"(-41/15, 31/15, 4/5, 5/36)","explanation_zh":"有序四則結果=(-41/15, 31/15, 4/5, 5/36)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有序四則運算結果"]},"variant":{"family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112012,"parameters":{"a":[-1,3],"b":[-12,5],"operation":"density11_arithmetic_ordered_four_results","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"-1/3\",\"-12/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0013","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"有序四則運算結果","stem_zh":"有理數Q；a=0/1、b=13/2。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。","answer":{"kind":"expression","value":"(13/2, -13/2, 0/1, 0/1)","explanation_zh":"有序四則結果=(13/2, -13/2, 0/1, 0/1)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有序四則運算結果"]},"variant":{"family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112013,"parameters":{"a":[0,1],"b":[13,2],"operation":"density11_arithmetic_ordered_four_results","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"0/1\",\"13/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0014","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"有序四則運算結果","stem_zh":"有理數Q；a=1/2、b=-14/3。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。","answer":{"kind":"expression","value":"(-25/6, 31/6, -7/3, -3/28)","explanation_zh":"有序四則結果=(-25/6, 31/6, -7/3, -3/28)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有序四則運算結果"]},"variant":{"family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112014,"parameters":{"a":[1,2],"b":[-14,3],"operation":"density11_arithmetic_ordered_four_results","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"1/2\",\"-14/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0015","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"有序四則運算結果","stem_zh":"有理數Q；a=2/3、b=15/4。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。","answer":{"kind":"expression","value":"(53/12, -37/12, 5/2, 8/45)","explanation_zh":"有序四則結果=(53/12, -37/12, 5/2, 8/45)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有序四則運算結果"]},"variant":{"family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112015,"parameters":{"a":[2,3],"b":[15,4],"operation":"density11_arithmetic_ordered_four_results","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"2/3\",\"15/4\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0016","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"有序四則運算結果","stem_zh":"有理數Q；a=3/1、b=-16/5。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。","answer":{"kind":"expression","value":"(-1/5, 31/5, -48/5, -15/16)","explanation_zh":"有序四則結果=(-1/5, 31/5, -48/5, -15/16)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有序四則運算結果"]},"variant":{"family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112016,"parameters":{"a":[3,1],"b":[-16,5],"operation":"density11_arithmetic_ordered_four_results","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"3/1\",\"-16/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0017","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"有序四則運算結果","stem_zh":"有理數Q；a=2/1、b=17/2。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。","answer":{"kind":"expression","value":"(21/2, -13/2, 17/1, 4/17)","explanation_zh":"有序四則結果=(21/2, -13/2, 17/1, 4/17)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["有序四則運算結果"]},"variant":{"family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112017,"parameters":{"a":[4,2],"b":[17,2],"operation":"density11_arithmetic_ordered_four_results","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"2/1\",\"17/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0018","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"有序四則運算結果","stem_zh":"有理數Q；a=5/3、b=-6/1。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。","answer":{"kind":"expression","value":"(-13/3, 23/3, -10/1, -5/18)","explanation_zh":"有序四則結果=(-13/3, 23/3, -10/1, -5/18)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有序四則運算結果"]},"variant":{"family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112018,"parameters":{"a":[5,3],"b":[-18,3],"operation":"density11_arithmetic_ordered_four_results","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"5/3\",\"-6/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0019","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"有序四則運算結果","stem_zh":"有理數Q；a=6/1、b=19/4。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。","answer":{"kind":"expression","value":"(43/4, 5/4, 57/2, 24/19)","explanation_zh":"有序四則結果=(43/4, 5/4, 57/2, 24/19)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有序四則運算結果"]},"variant":{"family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112019,"parameters":{"a":[6,1],"b":[19,4],"operation":"density11_arithmetic_ordered_four_results","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"6/1\",\"19/4\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0020","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"有序四則運算結果","stem_zh":"有理數Q；a=7/2、b=-4/1。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。","answer":{"kind":"expression","value":"(-1/2, 15/2, -14/1, -7/8)","explanation_zh":"有序四則結果=(-1/2, 15/2, -14/1, -7/8)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有序四則運算結果"]},"variant":{"family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112020,"parameters":{"a":[7,2],"b":[-20,5],"operation":"density11_arithmetic_ordered_four_results","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"7/2\",\"-4/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0021","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"有序四則運算結果","stem_zh":"有理數Q；a=8/3、b=21/2。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。","answer":{"kind":"expression","value":"(79/6, -47/6, 28/1, 16/63)","explanation_zh":"有序四則結果=(79/6, -47/6, 28/1, 16/63)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有序四則運算結果"]},"variant":{"family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112021,"parameters":{"a":[8,3],"b":[21,2],"operation":"density11_arithmetic_ordered_four_results","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"8/3\",\"21/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0022","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"有序四則運算結果","stem_zh":"有理數Q；a=9/1、b=-22/3。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。","answer":{"kind":"expression","value":"(5/3, 49/3, -66/1, -27/22)","explanation_zh":"有序四則結果=(5/3, 49/3, -66/1, -27/22)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有序四則運算結果"]},"variant":{"family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112022,"parameters":{"a":[9,1],"b":[-22,3],"operation":"density11_arithmetic_ordered_four_results","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"9/1\",\"-22/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0023","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"有序四則運算結果","stem_zh":"有理數Q；a=5/1、b=23/4。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。","answer":{"kind":"expression","value":"(43/4, -3/4, 115/4, 20/23)","explanation_zh":"有序四則結果=(43/4, -3/4, 115/4, 20/23)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有序四則運算結果"]},"variant":{"family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112023,"parameters":{"a":[10,2],"b":[23,4],"operation":"density11_arithmetic_ordered_four_results","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"5/1\",\"23/4\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0024","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"有序四則運算結果","stem_zh":"有理數Q；a=11/3、b=-24/5。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。","answer":{"kind":"expression","value":"(-17/15, 127/15, -88/5, -55/72)","explanation_zh":"有序四則結果=(-17/15, 127/15, -88/5, -55/72)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有序四則運算結果"]},"variant":{"family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112024,"parameters":{"a":[11,3],"b":[-24,5],"operation":"density11_arithmetic_ordered_four_results","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"11/3\",\"-24/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0025","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"有序四則運算結果","stem_zh":"有理數Q；a=12/1、b=25/2。依序求 (a+b, a−b, ab, a/b)，未定義的欄位寫未定義。","answer":{"kind":"expression","value":"(49/2, -1/2, 150/1, 24/25)","explanation_zh":"有序四則結果=(49/2, -1/2, 150/1, 24/25)。以正分母分數精確加減乘除，除零未定義；結果不使用自然數截斷運算。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["有序四則運算結果"]},"variant":{"family_id":"qf-density11-arithmetic-ordered-four-results","seed":202610112025,"parameters":{"a":[12,1],"b":[25,2],"operation":"density11_arithmetic_ordered_four_results","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-ordered-four-results/[\"12/1\",\"25/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0026","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"括號位置的雙式值","stem_zh":"有理數Q；a=0/1、b=2/1、c=0/1。依序求 ((a+b)c, a+bc)，保留括號。","answer":{"kind":"expression","value":"(0/1, 0/1)","explanation_zh":"括號雙式值=(0/1, 0/1)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["括號位置的雙式值"]},"variant":{"family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112026,"parameters":{"a":[0,1],"b":[2,1],"c":[0,1],"operation":"density11_arithmetic_parentheses_pair","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"zero_pair\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0027","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"括號位置的雙式值","stem_zh":"有理數Q；a=3/2、b=9/5、c=0/1。依序求 ((a+b)c, a+bc)，保留括號。","answer":{"kind":"expression","value":"(0/1, 3/2)","explanation_zh":"括號雙式值=(0/1, 3/2)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["括號位置的雙式值"]},"variant":{"family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112027,"parameters":{"a":[3,2],"b":[9,5],"c":[0,1],"operation":"density11_arithmetic_parentheses_pair","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"annihilated_outer\",\"3/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0028","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"括號位置的雙式值","stem_zh":"有理數Q；a=1/3、b=2/3、c=1/1。依序求 ((a+b)c, a+bc)，保留括號。","answer":{"kind":"expression","value":"(1/1, 1/1)","explanation_zh":"括號雙式值=(1/1, 1/1)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["括號位置的雙式值"]},"variant":{"family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112028,"parameters":{"a":[1,3],"b":[2,3],"c":[1,1],"operation":"density11_arithmetic_parentheses_pair","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"neutral_multiplier\",\"1/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0029","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"括號位置的雙式值","stem_zh":"有理數Q；a=0/1、b=2/3、c=3/4。依序求 ((a+b)c, a+bc)，保留括號。","answer":{"kind":"expression","value":"(1/2, 1/2)","explanation_zh":"括號雙式值=(1/2, 1/2)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["括號位置的雙式值"]},"variant":{"family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112029,"parameters":{"a":[0,1],"b":[2,3],"c":[3,4],"operation":"density11_arithmetic_parentheses_pair","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"equal_pair\",\"1/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0030","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"括號位置的雙式值","stem_zh":"有理數Q；a=-4/1、b=-1/1、c=5/2。依序求 ((a+b)c, a+bc)，保留括號。","answer":{"kind":"expression","value":"(-25/2, -13/2)","explanation_zh":"括號雙式值=(-25/2, -13/2)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["括號位置的雙式值"]},"variant":{"family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112030,"parameters":{"a":[-8,2],"b":[-3,3],"c":[5,2],"operation":"density11_arithmetic_parentheses_pair","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"-4/1\",\"-1/1\",\"5/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0031","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"括號位置的雙式值","stem_zh":"有理數Q；a=-7/2、b=-4/3、c=3/1。依序求 ((a+b)c, a+bc)，保留括號。","answer":{"kind":"expression","value":"(-29/2, -15/2)","explanation_zh":"括號雙式值=(-29/2, -15/2)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["括號位置的雙式值"]},"variant":{"family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112031,"parameters":{"a":[-7,2],"b":[-4,3],"c":[6,2],"operation":"density11_arithmetic_parentheses_pair","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"-7/2\",\"-4/3\",\"3/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0032","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"括號位置的雙式值","stem_zh":"有理數Q；a=-3/1、b=-5/3、c=7/2。依序求 ((a+b)c, a+bc)，保留括號。","answer":{"kind":"expression","value":"(-49/3, -53/6)","explanation_zh":"括號雙式值=(-49/3, -53/6)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["括號位置的雙式值"]},"variant":{"family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112032,"parameters":{"a":[-6,2],"b":[-5,3],"c":[7,2],"operation":"density11_arithmetic_parentheses_pair","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"-3/1\",\"-5/3\",\"7/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0033","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"括號位置的雙式值","stem_zh":"有理數Q；a=-5/2、b=-2/1、c=4/1。依序求 ((a+b)c, a+bc)，保留括號。","answer":{"kind":"expression","value":"(-18/1, -21/2)","explanation_zh":"括號雙式值=(-18/1, -21/2)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["括號位置的雙式值"]},"variant":{"family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112033,"parameters":{"a":[-5,2],"b":[-6,3],"c":[8,2],"operation":"density11_arithmetic_parentheses_pair","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"-5/2\",\"-2/1\",\"4/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0034","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"括號位置的雙式值","stem_zh":"有理數Q；a=-2/1、b=-7/3、c=9/2。依序求 ((a+b)c, a+bc)，保留括號。","answer":{"kind":"expression","value":"(-39/2, -25/2)","explanation_zh":"括號雙式值=(-39/2, -25/2)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["括號位置的雙式值"]},"variant":{"family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112034,"parameters":{"a":[-4,2],"b":[-7,3],"c":[9,2],"operation":"density11_arithmetic_parentheses_pair","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"-2/1\",\"-7/3\",\"9/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0035","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"括號位置的雙式值","stem_zh":"有理數Q；a=-3/2、b=-8/3、c=5/1。依序求 ((a+b)c, a+bc)，保留括號。","answer":{"kind":"expression","value":"(-125/6, -89/6)","explanation_zh":"括號雙式值=(-125/6, -89/6)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["括號位置的雙式值"]},"variant":{"family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112035,"parameters":{"a":[-3,2],"b":[-8,3],"c":[10,2],"operation":"density11_arithmetic_parentheses_pair","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"-3/2\",\"-8/3\",\"5/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0036","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"括號位置的雙式值","stem_zh":"有理數Q；a=-1/1、b=-3/1、c=11/2。依序求 ((a+b)c, a+bc)，保留括號。","answer":{"kind":"expression","value":"(-22/1, -35/2)","explanation_zh":"括號雙式值=(-22/1, -35/2)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["括號位置的雙式值"]},"variant":{"family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112036,"parameters":{"a":[-2,2],"b":[-9,3],"c":[11,2],"operation":"density11_arithmetic_parentheses_pair","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"-1/1\",\"-3/1\",\"11/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0037","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"括號位置的雙式值","stem_zh":"有理數Q；a=-1/2、b=-10/3、c=6/1。依序求 ((a+b)c, a+bc)，保留括號。","answer":{"kind":"expression","value":"(-23/1, -41/2)","explanation_zh":"括號雙式值=(-23/1, -41/2)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["括號位置的雙式值"]},"variant":{"family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112037,"parameters":{"a":[-1,2],"b":[-10,3],"c":[12,2],"operation":"density11_arithmetic_parentheses_pair","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"-1/2\",\"-10/3\",\"6/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0038","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"括號位置的雙式值","stem_zh":"有理數Q；a=0/1、b=-11/3、c=13/2。依序求 ((a+b)c, a+bc)，保留括號。","answer":{"kind":"expression","value":"(-143/6, -143/6)","explanation_zh":"括號雙式值=(-143/6, -143/6)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["括號位置的雙式值"]},"variant":{"family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112038,"parameters":{"a":[0,2],"b":[-11,3],"c":[13,2],"operation":"density11_arithmetic_parentheses_pair","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"equal_pair\",\"-143/6\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0039","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"括號位置的雙式值","stem_zh":"有理數Q；a=1/2、b=-4/1、c=7/1。依序求 ((a+b)c, a+bc)，保留括號。","answer":{"kind":"expression","value":"(-49/2, -55/2)","explanation_zh":"括號雙式值=(-49/2, -55/2)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["括號位置的雙式值"]},"variant":{"family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112039,"parameters":{"a":[1,2],"b":[-12,3],"c":[14,2],"operation":"density11_arithmetic_parentheses_pair","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"1/2\",\"-4/1\",\"7/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0040","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"括號位置的雙式值","stem_zh":"有理數Q；a=1/1、b=-13/3、c=15/2。依序求 ((a+b)c, a+bc)，保留括號。","answer":{"kind":"expression","value":"(-25/1, -63/2)","explanation_zh":"括號雙式值=(-25/1, -63/2)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["括號位置的雙式值"]},"variant":{"family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112040,"parameters":{"a":[2,2],"b":[-13,3],"c":[15,2],"operation":"density11_arithmetic_parentheses_pair","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"1/1\",\"-13/3\",\"15/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0041","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"括號位置的雙式值","stem_zh":"有理數Q；a=3/2、b=-14/3、c=8/1。依序求 ((a+b)c, a+bc)，保留括號。","answer":{"kind":"expression","value":"(-76/3, -215/6)","explanation_zh":"括號雙式值=(-76/3, -215/6)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["括號位置的雙式值"]},"variant":{"family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112041,"parameters":{"a":[3,2],"b":[-14,3],"c":[16,2],"operation":"density11_arithmetic_parentheses_pair","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"3/2\",\"-14/3\",\"8/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0042","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"括號位置的雙式值","stem_zh":"有理數Q；a=2/1、b=-5/1、c=17/2。依序求 ((a+b)c, a+bc)，保留括號。","answer":{"kind":"expression","value":"(-51/2, -81/2)","explanation_zh":"括號雙式值=(-51/2, -81/2)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["括號位置的雙式值"]},"variant":{"family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112042,"parameters":{"a":[4,2],"b":[-15,3],"c":[17,2],"operation":"density11_arithmetic_parentheses_pair","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"2/1\",\"-5/1\",\"17/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0043","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"括號位置的雙式值","stem_zh":"有理數Q；a=5/2、b=-16/3、c=9/1。依序求 ((a+b)c, a+bc)，保留括號。","answer":{"kind":"expression","value":"(-51/2, -91/2)","explanation_zh":"括號雙式值=(-51/2, -91/2)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["括號位置的雙式值"]},"variant":{"family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112043,"parameters":{"a":[5,2],"b":[-16,3],"c":[18,2],"operation":"density11_arithmetic_parentheses_pair","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"5/2\",\"-16/3\",\"9/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0044","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"括號位置的雙式值","stem_zh":"有理數Q；a=3/1、b=-17/3、c=19/2。依序求 ((a+b)c, a+bc)，保留括號。","answer":{"kind":"expression","value":"(-76/3, -305/6)","explanation_zh":"括號雙式值=(-76/3, -305/6)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["括號位置的雙式值"]},"variant":{"family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112044,"parameters":{"a":[6,2],"b":[-17,3],"c":[19,2],"operation":"density11_arithmetic_parentheses_pair","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"3/1\",\"-17/3\",\"19/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0045","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"括號位置的雙式值","stem_zh":"有理數Q；a=7/2、b=-6/1、c=10/1。依序求 ((a+b)c, a+bc)，保留括號。","answer":{"kind":"expression","value":"(-25/1, -113/2)","explanation_zh":"括號雙式值=(-25/1, -113/2)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["括號位置的雙式值"]},"variant":{"family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112045,"parameters":{"a":[7,2],"b":[-18,3],"c":[20,2],"operation":"density11_arithmetic_parentheses_pair","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"7/2\",\"-6/1\",\"10/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0046","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"括號位置的雙式值","stem_zh":"有理數Q；a=4/1、b=-19/3、c=21/2。依序求 ((a+b)c, a+bc)，保留括號。","answer":{"kind":"expression","value":"(-49/2, -125/2)","explanation_zh":"括號雙式值=(-49/2, -125/2)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["括號位置的雙式值"]},"variant":{"family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112046,"parameters":{"a":[8,2],"b":[-19,3],"c":[21,2],"operation":"density11_arithmetic_parentheses_pair","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"4/1\",\"-19/3\",\"21/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0047","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"括號位置的雙式值","stem_zh":"有理數Q；a=9/2、b=-20/3、c=11/1。依序求 ((a+b)c, a+bc)，保留括號。","answer":{"kind":"expression","value":"(-143/6, -413/6)","explanation_zh":"括號雙式值=(-143/6, -413/6)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["括號位置的雙式值"]},"variant":{"family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112047,"parameters":{"a":[9,2],"b":[-20,3],"c":[22,2],"operation":"density11_arithmetic_parentheses_pair","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"9/2\",\"-20/3\",\"11/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0048","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"括號位置的雙式值","stem_zh":"有理數Q；a=5/1、b=-7/1、c=23/2。依序求 ((a+b)c, a+bc)，保留括號。","answer":{"kind":"expression","value":"(-23/1, -151/2)","explanation_zh":"括號雙式值=(-23/1, -151/2)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["括號位置的雙式值"]},"variant":{"family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112048,"parameters":{"a":[10,2],"b":[-21,3],"c":[23,2],"operation":"density11_arithmetic_parentheses_pair","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"5/1\",\"-7/1\",\"23/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0049","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"括號位置的雙式值","stem_zh":"有理數Q；a=11/2、b=-22/3、c=12/1。依序求 ((a+b)c, a+bc)，保留括號。","answer":{"kind":"expression","value":"(-22/1, -165/2)","explanation_zh":"括號雙式值=(-22/1, -165/2)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["括號位置的雙式值"]},"variant":{"family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112049,"parameters":{"a":[11,2],"b":[-22,3],"c":[24,2],"operation":"density11_arithmetic_parentheses_pair","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"11/2\",\"-22/3\",\"12/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0050","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"括號位置的雙式值","stem_zh":"有理數Q；a=6/1、b=-23/3、c=25/2。依序求 ((a+b)c, a+bc)，保留括號。","answer":{"kind":"expression","value":"(-125/6, -539/6)","explanation_zh":"括號雙式值=(-125/6, -539/6)。第一式先加後乘，第二式先乘後加；零乘數與一乘數可使某些項消失，不因此改計算次序。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["括號位置的雙式值"]},"variant":{"family_id":"qf-density11-arithmetic-parentheses-pair","seed":202610112050,"parameters":{"a":[12,2],"b":[-23,3],"c":[25,2],"operation":"density11_arithmetic_parentheses_pair","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-parentheses-pair/[\"6/1\",\"-23/3\",\"25/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0051","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"乘法逆元資格與乘積缺陷","stem_zh":"有理數Q；a=0/1、候選r=7/2。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。","answer":{"kind":"expression","value":"(否, -1/1)","explanation_zh":"逆元資格與乘積缺陷=(否, -1/1)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["乘法逆元資格與乘積缺陷"]},"variant":{"family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112051,"parameters":{"a":[0,1],"r":[7,2],"operation":"density11_arithmetic_inverse_candidate","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"zero_operand\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0052","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"乘法逆元資格與乘積缺陷","stem_zh":"有理數Q；a=2/3、候選r=0/1。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。","answer":{"kind":"expression","value":"(否, -1/1)","explanation_zh":"逆元資格與乘積缺陷=(否, -1/1)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["乘法逆元資格與乘積缺陷"]},"variant":{"family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112052,"parameters":{"a":[2,3],"r":[0,1],"operation":"density11_arithmetic_inverse_candidate","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"zero_candidate\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0053","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"乘法逆元資格與乘積缺陷","stem_zh":"有理數Q；a=3/2、候選r=1/4。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。","answer":{"kind":"expression","value":"(否, -5/8)","explanation_zh":"逆元資格與乘積缺陷=(否, -5/8)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["乘法逆元資格與乘積缺陷"]},"variant":{"family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112053,"parameters":{"a":[3,2],"r":[1,4],"operation":"density11_arithmetic_inverse_candidate","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"1/4\",\"3/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0054","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"乘法逆元資格與乘積缺陷","stem_zh":"有理數Q；a=-2/1、候選r=-1/2。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。","answer":{"kind":"expression","value":"(是, 0/1)","explanation_zh":"逆元資格與乘積缺陷=(是, 0/1)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["乘法逆元資格與乘積缺陷"]},"variant":{"family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112054,"parameters":{"a":[-4,2],"r":[-2,4],"operation":"density11_arithmetic_inverse_candidate","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"-1/2\",\"-2/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0055","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"乘法逆元資格與乘積缺陷","stem_zh":"有理數Q；a=5/2、候選r=1/6。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。","answer":{"kind":"expression","value":"(否, -7/12)","explanation_zh":"逆元資格與乘積缺陷=(否, -7/12)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["乘法逆元資格與乘積缺陷"]},"variant":{"family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112055,"parameters":{"a":[5,2],"r":[1,6],"operation":"density11_arithmetic_inverse_candidate","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"1/6\",\"5/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0056","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"乘法逆元資格與乘積缺陷","stem_zh":"有理數Q；a=-3/1、候選r=1/7。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。","answer":{"kind":"expression","value":"(否, -10/7)","explanation_zh":"逆元資格與乘積缺陷=(否, -10/7)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["乘法逆元資格與乘積缺陷"]},"variant":{"family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112056,"parameters":{"a":[-6,2],"r":[1,7],"operation":"density11_arithmetic_inverse_candidate","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"-3/1\",\"1/7\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0057","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"乘法逆元資格與乘積缺陷","stem_zh":"有理數Q；a=7/2、候選r=2/7。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。","answer":{"kind":"expression","value":"(是, 0/1)","explanation_zh":"逆元資格與乘積缺陷=(是, 0/1)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["乘法逆元資格與乘積缺陷"]},"variant":{"family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112057,"parameters":{"a":[7,2],"r":[2,7],"operation":"density11_arithmetic_inverse_candidate","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"2/7\",\"7/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0058","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"乘法逆元資格與乘積缺陷","stem_zh":"有理數Q；a=-4/1、候選r=1/9。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。","answer":{"kind":"expression","value":"(否, -13/9)","explanation_zh":"逆元資格與乘積缺陷=(否, -13/9)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["乘法逆元資格與乘積缺陷"]},"variant":{"family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112058,"parameters":{"a":[-8,2],"r":[1,9],"operation":"density11_arithmetic_inverse_candidate","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"-4/1\",\"1/9\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0059","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"乘法逆元資格與乘積缺陷","stem_zh":"有理數Q；a=9/2、候選r=1/10。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。","answer":{"kind":"expression","value":"(否, -11/20)","explanation_zh":"逆元資格與乘積缺陷=(否, -11/20)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["乘法逆元資格與乘積缺陷"]},"variant":{"family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112059,"parameters":{"a":[9,2],"r":[1,10],"operation":"density11_arithmetic_inverse_candidate","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"1/10\",\"9/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0060","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"乘法逆元資格與乘積缺陷","stem_zh":"有理數Q；a=-5/1、候選r=-1/5。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。","answer":{"kind":"expression","value":"(是, 0/1)","explanation_zh":"逆元資格與乘積缺陷=(是, 0/1)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["乘法逆元資格與乘積缺陷"]},"variant":{"family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112060,"parameters":{"a":[-10,2],"r":[-2,10],"operation":"density11_arithmetic_inverse_candidate","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"-1/5\",\"-5/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0061","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"乘法逆元資格與乘積缺陷","stem_zh":"有理數Q；a=11/2、候選r=1/12。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。","answer":{"kind":"expression","value":"(否, -13/24)","explanation_zh":"逆元資格與乘積缺陷=(否, -13/24)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["乘法逆元資格與乘積缺陷"]},"variant":{"family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112061,"parameters":{"a":[11,2],"r":[1,12],"operation":"density11_arithmetic_inverse_candidate","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"1/12\",\"11/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0062","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"乘法逆元資格與乘積缺陷","stem_zh":"有理數Q；a=-6/1、候選r=1/13。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。","answer":{"kind":"expression","value":"(否, -19/13)","explanation_zh":"逆元資格與乘積缺陷=(否, -19/13)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["乘法逆元資格與乘積缺陷"]},"variant":{"family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112062,"parameters":{"a":[-12,2],"r":[1,13],"operation":"density11_arithmetic_inverse_candidate","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"-6/1\",\"1/13\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0063","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"乘法逆元資格與乘積缺陷","stem_zh":"有理數Q；a=13/2、候選r=2/13。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。","answer":{"kind":"expression","value":"(是, 0/1)","explanation_zh":"逆元資格與乘積缺陷=(是, 0/1)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["乘法逆元資格與乘積缺陷"]},"variant":{"family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112063,"parameters":{"a":[13,2],"r":[2,13],"operation":"density11_arithmetic_inverse_candidate","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"13/2\",\"2/13\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0064","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"乘法逆元資格與乘積缺陷","stem_zh":"有理數Q；a=-7/1、候選r=1/15。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。","answer":{"kind":"expression","value":"(否, -22/15)","explanation_zh":"逆元資格與乘積缺陷=(否, -22/15)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["乘法逆元資格與乘積缺陷"]},"variant":{"family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112064,"parameters":{"a":[-14,2],"r":[1,15],"operation":"density11_arithmetic_inverse_candidate","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"-7/1\",\"1/15\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0065","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"乘法逆元資格與乘積缺陷","stem_zh":"有理數Q；a=15/2、候選r=1/16。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。","answer":{"kind":"expression","value":"(否, -17/32)","explanation_zh":"逆元資格與乘積缺陷=(否, -17/32)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["乘法逆元資格與乘積缺陷"]},"variant":{"family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112065,"parameters":{"a":[15,2],"r":[1,16],"operation":"density11_arithmetic_inverse_candidate","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"1/16\",\"15/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0066","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"乘法逆元資格與乘積缺陷","stem_zh":"有理數Q；a=-8/1、候選r=-1/8。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。","answer":{"kind":"expression","value":"(是, 0/1)","explanation_zh":"逆元資格與乘積缺陷=(是, 0/1)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["乘法逆元資格與乘積缺陷"]},"variant":{"family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112066,"parameters":{"a":[-16,2],"r":[-2,16],"operation":"density11_arithmetic_inverse_candidate","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"-1/8\",\"-8/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0067","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"乘法逆元資格與乘積缺陷","stem_zh":"有理數Q；a=17/2、候選r=1/18。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。","answer":{"kind":"expression","value":"(否, -19/36)","explanation_zh":"逆元資格與乘積缺陷=(否, -19/36)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["乘法逆元資格與乘積缺陷"]},"variant":{"family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112067,"parameters":{"a":[17,2],"r":[1,18],"operation":"density11_arithmetic_inverse_candidate","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"1/18\",\"17/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0068","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"乘法逆元資格與乘積缺陷","stem_zh":"有理數Q；a=-9/1、候選r=1/19。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。","answer":{"kind":"expression","value":"(否, -28/19)","explanation_zh":"逆元資格與乘積缺陷=(否, -28/19)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["乘法逆元資格與乘積缺陷"]},"variant":{"family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112068,"parameters":{"a":[-18,2],"r":[1,19],"operation":"density11_arithmetic_inverse_candidate","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"-9/1\",\"1/19\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0069","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"乘法逆元資格與乘積缺陷","stem_zh":"有理數Q；a=19/2、候選r=2/19。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。","answer":{"kind":"expression","value":"(是, 0/1)","explanation_zh":"逆元資格與乘積缺陷=(是, 0/1)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["乘法逆元資格與乘積缺陷"]},"variant":{"family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112069,"parameters":{"a":[19,2],"r":[2,19],"operation":"density11_arithmetic_inverse_candidate","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"19/2\",\"2/19\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0070","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"乘法逆元資格與乘積缺陷","stem_zh":"有理數Q；a=-10/1、候選r=1/21。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。","answer":{"kind":"expression","value":"(否, -31/21)","explanation_zh":"逆元資格與乘積缺陷=(否, -31/21)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["乘法逆元資格與乘積缺陷"]},"variant":{"family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112070,"parameters":{"a":[-20,2],"r":[1,21],"operation":"density11_arithmetic_inverse_candidate","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"-10/1\",\"1/21\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0071","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"乘法逆元資格與乘積缺陷","stem_zh":"有理數Q；a=21/2、候選r=1/22。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。","answer":{"kind":"expression","value":"(否, -23/44)","explanation_zh":"逆元資格與乘積缺陷=(否, -23/44)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["乘法逆元資格與乘積缺陷"]},"variant":{"family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112071,"parameters":{"a":[21,2],"r":[1,22],"operation":"density11_arithmetic_inverse_candidate","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"1/22\",\"21/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0072","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"乘法逆元資格與乘積缺陷","stem_zh":"有理數Q；a=-11/1、候選r=-1/11。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。","answer":{"kind":"expression","value":"(是, 0/1)","explanation_zh":"逆元資格與乘積缺陷=(是, 0/1)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["乘法逆元資格與乘積缺陷"]},"variant":{"family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112072,"parameters":{"a":[-22,2],"r":[-2,22],"operation":"density11_arithmetic_inverse_candidate","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"-1/11\",\"-11/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0073","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"乘法逆元資格與乘積缺陷","stem_zh":"有理數Q；a=23/2、候選r=1/24。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。","answer":{"kind":"expression","value":"(否, -25/48)","explanation_zh":"逆元資格與乘積缺陷=(否, -25/48)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["乘法逆元資格與乘積缺陷"]},"variant":{"family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112073,"parameters":{"a":[23,2],"r":[1,24],"operation":"density11_arithmetic_inverse_candidate","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"1/24\",\"23/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0074","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"乘法逆元資格與乘積缺陷","stem_zh":"有理數Q；a=-12/1、候選r=1/25。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。","answer":{"kind":"expression","value":"(否, -37/25)","explanation_zh":"逆元資格與乘積缺陷=(否, -37/25)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["乘法逆元資格與乘積缺陷"]},"variant":{"family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112074,"parameters":{"a":[-24,2],"r":[1,25],"operation":"density11_arithmetic_inverse_candidate","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"-12/1\",\"1/25\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0075","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"乘法逆元資格與乘積缺陷","stem_zh":"有理數Q；a=25/2、候選r=2/25。依序答 (r是否為a的乘法逆元, ar−1)，第二欄以精確分數表示。","answer":{"kind":"expression","value":"(是, 0/1)","explanation_zh":"逆元資格與乘積缺陷=(是, 0/1)。零沒有乘法逆元；非零a也須乘積恰為一才通過，缺陷保留正負號。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["乘法逆元資格與乘積缺陷"]},"variant":{"family_id":"qf-density11-arithmetic-inverse-candidate","seed":202610112075,"parameters":{"a":[25,2],"r":[2,25],"operation":"density11_arithmetic_inverse_candidate","domain":"rational_arithmetic_slice","semantic_template":"qf-density11-arithmetic-inverse-candidate/[\"2/25\",\"25/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0076","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"Q運算結果與自然數封閉資格","stem_zh":"有理數Q運算，N包含0；自然數輸入a=0、b=0。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。","answer":{"kind":"expression","value":"(0, 是, 未定義, 未定義)","explanation_zh":"Q結果與N封閉資格=(0, 是, 未定義, 未定義)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["Q運算結果與自然數封閉資格"]},"variant":{"family_id":"qf-density11-arithmetic-system-membership","seed":202610112076,"parameters":{"a":0,"b":0,"operation":"density11_arithmetic_system_membership","domain":"N_operands_Q_results","semantic_template":"qf-density11-arithmetic-system-membership/[0,0]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0077","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"Q運算結果與自然數封閉資格","stem_zh":"有理數Q運算，N包含0；自然數輸入a=1、b=0。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。","answer":{"kind":"expression","value":"(1, 是, 未定義, 未定義)","explanation_zh":"Q結果與N封閉資格=(1, 是, 未定義, 未定義)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["Q運算結果與自然數封閉資格"]},"variant":{"family_id":"qf-density11-arithmetic-system-membership","seed":202610112077,"parameters":{"a":1,"b":0,"operation":"density11_arithmetic_system_membership","domain":"N_operands_Q_results","semantic_template":"qf-density11-arithmetic-system-membership/[1,0]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0078","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"Q運算結果與自然數封閉資格","stem_zh":"有理數Q運算，N包含0；自然數輸入a=1、b=2。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。","answer":{"kind":"expression","value":"(-1, 否, 1/2, 否)","explanation_zh":"Q結果與N封閉資格=(-1, 否, 1/2, 否)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["Q運算結果與自然數封閉資格"]},"variant":{"family_id":"qf-density11-arithmetic-system-membership","seed":202610112078,"parameters":{"a":1,"b":2,"operation":"density11_arithmetic_system_membership","domain":"N_operands_Q_results","semantic_template":"qf-density11-arithmetic-system-membership/[1,2]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0079","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"Q運算結果與自然數封閉資格","stem_zh":"有理數Q運算，N包含0；自然數輸入a=6、b=3。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。","answer":{"kind":"expression","value":"(3, 是, 2/1, 是)","explanation_zh":"Q結果與N封閉資格=(3, 是, 2/1, 是)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["Q運算結果與自然數封閉資格"]},"variant":{"family_id":"qf-density11-arithmetic-system-membership","seed":202610112079,"parameters":{"a":6,"b":3,"operation":"density11_arithmetic_system_membership","domain":"N_operands_Q_results","semantic_template":"qf-density11-arithmetic-system-membership/[6,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0080","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"Q運算結果與自然數封閉資格","stem_zh":"有理數Q運算，N包含0；自然數輸入a=0、b=3。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。","answer":{"kind":"expression","value":"(-3, 否, 0/1, 是)","explanation_zh":"Q結果與N封閉資格=(-3, 否, 0/1, 是)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["Q運算結果與自然數封閉資格"]},"variant":{"family_id":"qf-density11-arithmetic-system-membership","seed":202610112080,"parameters":{"a":0,"b":3,"operation":"density11_arithmetic_system_membership","domain":"N_operands_Q_results","semantic_template":"qf-density11-arithmetic-system-membership/[0,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0081","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"Q運算結果與自然數封閉資格","stem_zh":"有理數Q運算，N包含0；自然數輸入a=2、b=1。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。","answer":{"kind":"expression","value":"(1, 是, 2/1, 是)","explanation_zh":"Q結果與N封閉資格=(1, 是, 2/1, 是)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["Q運算結果與自然數封閉資格"]},"variant":{"family_id":"qf-density11-arithmetic-system-membership","seed":202610112081,"parameters":{"a":2,"b":1,"operation":"density11_arithmetic_system_membership","domain":"N_operands_Q_results","semantic_template":"qf-density11-arithmetic-system-membership/[2,1]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0082","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"Q運算結果與自然數封閉資格","stem_zh":"有理數Q運算，N包含0；自然數輸入a=3、b=2。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。","answer":{"kind":"expression","value":"(1, 是, 3/2, 否)","explanation_zh":"Q結果與N封閉資格=(1, 是, 3/2, 否)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["Q運算結果與自然數封閉資格"]},"variant":{"family_id":"qf-density11-arithmetic-system-membership","seed":202610112082,"parameters":{"a":3,"b":2,"operation":"density11_arithmetic_system_membership","domain":"N_operands_Q_results","semantic_template":"qf-density11-arithmetic-system-membership/[3,2]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0083","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"Q運算結果與自然數封閉資格","stem_zh":"有理數Q運算，N包含0；自然數輸入a=4、b=3。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。","answer":{"kind":"expression","value":"(1, 是, 4/3, 否)","explanation_zh":"Q結果與N封閉資格=(1, 是, 4/3, 否)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"},"difficulty":{"band":"lower_secondary","level":1,"skills":["Q運算結果與自然數封閉資格"]},"variant":{"family_id":"qf-density11-arithmetic-system-membership","seed":202610112083,"parameters":{"a":4,"b":3,"operation":"density11_arithmetic_system_membership","domain":"N_operands_Q_results","semantic_template":"qf-density11-arithmetic-system-membership/[4,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0084","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"Q運算結果與自然數封閉資格","stem_zh":"有理數Q運算，N包含0；自然數輸入a=5、b=4。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。","answer":{"kind":"expression","value":"(1, 是, 5/4, 否)","explanation_zh":"Q結果與N封閉資格=(1, 是, 5/4, 否)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["Q運算結果與自然數封閉資格"]},"variant":{"family_id":"qf-density11-arithmetic-system-membership","seed":202610112084,"parameters":{"a":5,"b":4,"operation":"density11_arithmetic_system_membership","domain":"N_operands_Q_results","semantic_template":"qf-density11-arithmetic-system-membership/[5,4]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0085","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"Q運算結果與自然數封閉資格","stem_zh":"有理數Q運算，N包含0；自然數輸入a=6、b=5。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。","answer":{"kind":"expression","value":"(1, 是, 6/5, 否)","explanation_zh":"Q結果與N封閉資格=(1, 是, 6/5, 否)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["Q運算結果與自然數封閉資格"]},"variant":{"family_id":"qf-density11-arithmetic-system-membership","seed":202610112085,"parameters":{"a":6,"b":5,"operation":"density11_arithmetic_system_membership","domain":"N_operands_Q_results","semantic_template":"qf-density11-arithmetic-system-membership/[6,5]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0086","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"Q運算結果與自然數封閉資格","stem_zh":"有理數Q運算，N包含0；自然數輸入a=7、b=1。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。","answer":{"kind":"expression","value":"(6, 是, 7/1, 是)","explanation_zh":"Q結果與N封閉資格=(6, 是, 7/1, 是)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["Q運算結果與自然數封閉資格"]},"variant":{"family_id":"qf-density11-arithmetic-system-membership","seed":202610112086,"parameters":{"a":7,"b":1,"operation":"density11_arithmetic_system_membership","domain":"N_operands_Q_results","semantic_template":"qf-density11-arithmetic-system-membership/[7,1]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0087","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"Q運算結果與自然數封閉資格","stem_zh":"有理數Q運算，N包含0；自然數輸入a=8、b=2。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。","answer":{"kind":"expression","value":"(6, 是, 4/1, 是)","explanation_zh":"Q結果與N封閉資格=(6, 是, 4/1, 是)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["Q運算結果與自然數封閉資格"]},"variant":{"family_id":"qf-density11-arithmetic-system-membership","seed":202610112087,"parameters":{"a":8,"b":2,"operation":"density11_arithmetic_system_membership","domain":"N_operands_Q_results","semantic_template":"qf-density11-arithmetic-system-membership/[8,2]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0088","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"Q運算結果與自然數封閉資格","stem_zh":"有理數Q運算，N包含0；自然數輸入a=9、b=3。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。","answer":{"kind":"expression","value":"(6, 是, 3/1, 是)","explanation_zh":"Q結果與N封閉資格=(6, 是, 3/1, 是)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["Q運算結果與自然數封閉資格"]},"variant":{"family_id":"qf-density11-arithmetic-system-membership","seed":202610112088,"parameters":{"a":9,"b":3,"operation":"density11_arithmetic_system_membership","domain":"N_operands_Q_results","semantic_template":"qf-density11-arithmetic-system-membership/[9,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0089","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"Q運算結果與自然數封閉資格","stem_zh":"有理數Q運算，N包含0；自然數輸入a=10、b=4。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。","answer":{"kind":"expression","value":"(6, 是, 5/2, 否)","explanation_zh":"Q結果與N封閉資格=(6, 是, 5/2, 否)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["Q運算結果與自然數封閉資格"]},"variant":{"family_id":"qf-density11-arithmetic-system-membership","seed":202610112089,"parameters":{"a":10,"b":4,"operation":"density11_arithmetic_system_membership","domain":"N_operands_Q_results","semantic_template":"qf-density11-arithmetic-system-membership/[10,4]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0090","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"Q運算結果與自然數封閉資格","stem_zh":"有理數Q運算，N包含0；自然數輸入a=11、b=5。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。","answer":{"kind":"expression","value":"(6, 是, 11/5, 否)","explanation_zh":"Q結果與N封閉資格=(6, 是, 11/5, 否)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["Q運算結果與自然數封閉資格"]},"variant":{"family_id":"qf-density11-arithmetic-system-membership","seed":202610112090,"parameters":{"a":11,"b":5,"operation":"density11_arithmetic_system_membership","domain":"N_operands_Q_results","semantic_template":"qf-density11-arithmetic-system-membership/[11,5]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0091","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"Q運算結果與自然數封閉資格","stem_zh":"有理數Q運算，N包含0；自然數輸入a=12、b=1。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。","answer":{"kind":"expression","value":"(11, 是, 12/1, 是)","explanation_zh":"Q結果與N封閉資格=(11, 是, 12/1, 是)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["Q運算結果與自然數封閉資格"]},"variant":{"family_id":"qf-density11-arithmetic-system-membership","seed":202610112091,"parameters":{"a":12,"b":1,"operation":"density11_arithmetic_system_membership","domain":"N_operands_Q_results","semantic_template":"qf-density11-arithmetic-system-membership/[12,1]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0092","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"Q運算結果與自然數封閉資格","stem_zh":"有理數Q運算，N包含0；自然數輸入a=13、b=2。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。","answer":{"kind":"expression","value":"(11, 是, 13/2, 否)","explanation_zh":"Q結果與N封閉資格=(11, 是, 13/2, 否)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"},"difficulty":{"band":"lower_secondary","level":2,"skills":["Q運算結果與自然數封閉資格"]},"variant":{"family_id":"qf-density11-arithmetic-system-membership","seed":202610112092,"parameters":{"a":13,"b":2,"operation":"density11_arithmetic_system_membership","domain":"N_operands_Q_results","semantic_template":"qf-density11-arithmetic-system-membership/[13,2]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0093","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"Q運算結果與自然數封閉資格","stem_zh":"有理數Q運算，N包含0；自然數輸入a=14、b=3。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。","answer":{"kind":"expression","value":"(11, 是, 14/3, 否)","explanation_zh":"Q結果與N封閉資格=(11, 是, 14/3, 否)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["Q運算結果與自然數封閉資格"]},"variant":{"family_id":"qf-density11-arithmetic-system-membership","seed":202610112093,"parameters":{"a":14,"b":3,"operation":"density11_arithmetic_system_membership","domain":"N_operands_Q_results","semantic_template":"qf-density11-arithmetic-system-membership/[14,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0094","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"Q運算結果與自然數封閉資格","stem_zh":"有理數Q運算，N包含0；自然數輸入a=15、b=4。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。","answer":{"kind":"expression","value":"(11, 是, 15/4, 否)","explanation_zh":"Q結果與N封閉資格=(11, 是, 15/4, 否)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["Q運算結果與自然數封閉資格"]},"variant":{"family_id":"qf-density11-arithmetic-system-membership","seed":202610112094,"parameters":{"a":15,"b":4,"operation":"density11_arithmetic_system_membership","domain":"N_operands_Q_results","semantic_template":"qf-density11-arithmetic-system-membership/[15,4]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0095","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"Q運算結果與自然數封閉資格","stem_zh":"有理數Q運算，N包含0；自然數輸入a=16、b=5。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。","answer":{"kind":"expression","value":"(11, 是, 16/5, 否)","explanation_zh":"Q結果與N封閉資格=(11, 是, 16/5, 否)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["Q運算結果與自然數封閉資格"]},"variant":{"family_id":"qf-density11-arithmetic-system-membership","seed":202610112095,"parameters":{"a":16,"b":5,"operation":"density11_arithmetic_system_membership","domain":"N_operands_Q_results","semantic_template":"qf-density11-arithmetic-system-membership/[16,5]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0096","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"Q運算結果與自然數封閉資格","stem_zh":"有理數Q運算，N包含0；自然數輸入a=17、b=1。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。","answer":{"kind":"expression","value":"(16, 是, 17/1, 是)","explanation_zh":"Q結果與N封閉資格=(16, 是, 17/1, 是)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["Q運算結果與自然數封閉資格"]},"variant":{"family_id":"qf-density11-arithmetic-system-membership","seed":202610112096,"parameters":{"a":17,"b":1,"operation":"density11_arithmetic_system_membership","domain":"N_operands_Q_results","semantic_template":"qf-density11-arithmetic-system-membership/[17,1]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0097","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"Q運算結果與自然數封閉資格","stem_zh":"有理數Q運算，N包含0；自然數輸入a=18、b=2。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。","answer":{"kind":"expression","value":"(16, 是, 9/1, 是)","explanation_zh":"Q結果與N封閉資格=(16, 是, 9/1, 是)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["Q運算結果與自然數封閉資格"]},"variant":{"family_id":"qf-density11-arithmetic-system-membership","seed":202610112097,"parameters":{"a":18,"b":2,"operation":"density11_arithmetic_system_membership","domain":"N_operands_Q_results","semantic_template":"qf-density11-arithmetic-system-membership/[18,2]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0098","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"Q運算結果與自然數封閉資格","stem_zh":"有理數Q運算，N包含0；自然數輸入a=19、b=3。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。","answer":{"kind":"expression","value":"(16, 是, 19/3, 否)","explanation_zh":"Q結果與N封閉資格=(16, 是, 19/3, 否)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["Q運算結果與自然數封閉資格"]},"variant":{"family_id":"qf-density11-arithmetic-system-membership","seed":202610112098,"parameters":{"a":19,"b":3,"operation":"density11_arithmetic_system_membership","domain":"N_operands_Q_results","semantic_template":"qf-density11-arithmetic-system-membership/[19,3]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0099","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"Q運算結果與自然數封閉資格","stem_zh":"有理數Q運算，N包含0；自然數輸入a=20、b=4。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。","answer":{"kind":"expression","value":"(16, 是, 5/1, 是)","explanation_zh":"Q結果與N封閉資格=(16, 是, 5/1, 是)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["Q運算結果與自然數封閉資格"]},"variant":{"family_id":"qf-density11-arithmetic-system-membership","seed":202610112099,"parameters":{"a":20,"b":4,"operation":"density11_arithmetic_system_membership","domain":"N_operands_Q_results","semantic_template":"qf-density11-arithmetic-system-membership/[20,4]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261011-r02-arithmetic-operations-0100","batch_id":"batch-20261008-advance-20261011-r02-arithmetic-operations-foundations","language":"zh-Hant","source_mko_ids":["mko-arithmetic-operations"],"title_zh":"Q運算結果與自然數封閉資格","stem_zh":"有理數Q運算，N包含0；自然數輸入a=21、b=5。依序求 (a−b, 差是否屬N, a/b, 商是否屬N)；商或其資格未定義則寫未定義。","answer":{"kind":"expression","value":"(16, 是, 21/5, 否)","explanation_zh":"Q結果與N封閉資格=(16, 是, 21/5, 否)。使用精確Q結果，非截斷減法或整除；負差不屬N，非整數商不屬N，除零未定義。"},"difficulty":{"band":"lower_secondary","level":3,"skills":["Q運算結果與自然數封閉資格"]},"variant":{"family_id":"qf-density11-arithmetic-system-membership","seed":202610112100,"parameters":{"a":21,"b":5,"operation":"density11_arithmetic_system_membership","domain":"N_operands_Q_results","semantic_template":"qf-density11-arithmetic-system-membership/[21,5]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}