Explore related knowledge and questions →
← Question derivation index · Original question batch
R0011-06163 · q-20261009-advance-20261013-r02-integer-0088
Ordered representations=[0/1, 24/-2, 13/27, -14/-29] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.
G001 — Ordered representations=[0/1, 24/-2, 13/27, -14/-29] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
declared exact integer or rational-representation protocol
After
retain signs,ordered operands and original index order
Conditions: none
Check these finite data with exact integer arithmetic,without floating approximations or a claim to prove all integer-ring axioms.
Rule: integer_arithmetic_scope · Uses: G001
Before
position=0;raw fraction=[0, 1]
After
normalized=0/1;integer=Yes
Conditions: none
A nonzero negative denominator is valid.Normalize to a reduced value with positive denominator;denominator1 means integer.Preserve original relative order within each group.
Rule: normalized_rational_integer_qualification · Uses: G001, S001
Before
position=1;raw fraction=[24, -2]
After
normalized=-12/1;integer=Yes
Conditions: none
A nonzero negative denominator is valid.Normalize to a reduced value with positive denominator;denominator1 means integer.Preserve original relative order within each group.
Rule: normalized_rational_integer_qualification · Uses: G001, S002
Before
position=2;raw fraction=[13, 27]
After
normalized=13/27;integer=No
Conditions: none
A nonzero negative denominator is valid.Normalize to a reduced value with positive denominator;denominator1 means integer.Preserve original relative order within each group.
Rule: normalized_rational_integer_qualification · Uses: G001, S003
Before
position=3;raw fraction=[-14, -29]
After
normalized=14/29;integer=No
Conditions: none
A nonzero negative denominator is valid.Normalize to a reduced value with positive denominator;denominator1 means integer.Preserve original relative order within each group.
Rule: normalized_rational_integer_qualification · Uses: G001, S004
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.([Yes, Yes, No, No], [0, -12], [13/27, 14/29])
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: c9b1599c3f394beafaa6c3f12687ae2e10aca811cafc245b6aa84f88af7599ffcandidate record SHA-256: acb0e83f7fcd8c97a8edaff187b58e15ca81ebbe5477a30594265a3541cf9ca0semantic review receipt SHA-256: b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80question-first witness SHA-256: 93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55Lean: NOT_RUN · human review: NOT_PERFORMED