Exact Rational Operations and Natural Eligibility
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261008-advance-20261011-r02-arithmetic-operations-foundations · 2026-10-08 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Ordered Four Rational Operations
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=1/2, b=0/1. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 1/2, 0/1, undefined)
Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (1/2, 1/2, 0/1, undefined).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-ordered-four-results · candidate
2. Ordered Four Rational Operations
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=0/1, b=-2/3. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-2/3, 2/3, 0/1, 0/1)
Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-2/3, 2/3, 0/1, 0/1).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-ordered-four-results · candidate
3. Ordered Four Rational Operations
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-10/3, b=3/4. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-31/12, -49/12, -5/2, -40/9)
Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-31/12, -49/12, -5/2, -40/9).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-ordered-four-results · candidate
4. Ordered Four Rational Operations
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-9/1, b=-4/5. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-49/5, -41/5, 36/5, 45/4)
Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-49/5, -41/5, 36/5, 45/4).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-ordered-four-results · candidate
5. Ordered Four Rational Operations
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-8/2, b=5/2. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-3/2, -13/2, -10/1, -8/5)
Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-3/2, -13/2, -10/1, -8/5).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-ordered-four-results · candidate
6. Ordered Four Rational Operations
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-7/3, b=-6/3. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-13/3, -1/3, 14/3, 7/6)
Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-13/3, -1/3, 14/3, 7/6).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-ordered-four-results · candidate
7. Ordered Four Rational Operations
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-6/1, b=7/4. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-17/4, -31/4, -21/2, -24/7)
Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-17/4, -31/4, -21/2, -24/7).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-ordered-four-results · candidate
8. Ordered Four Rational Operations
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-5/2, b=-8/5. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-41/10, -9/10, 4/1, 25/16)
Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-41/10, -9/10, 4/1, 25/16).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-ordered-four-results · candidate
9. Ordered Four Rational Operations
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-4/3, b=9/2. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(19/6, -35/6, -6/1, -8/27)
Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (19/6, -35/6, -6/1, -8/27).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-ordered-four-results · candidate
10. Ordered Four Rational Operations
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-3/1, b=-10/3. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-19/3, 1/3, 10/1, 9/10)
Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-19/3, 1/3, 10/1, 9/10).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-ordered-four-results · candidate
11. Ordered Four Rational Operations
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-2/2, b=11/4. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(7/4, -15/4, -11/4, -4/11)
Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (7/4, -15/4, -11/4, -4/11).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-ordered-four-results · candidate
12. Ordered Four Rational Operations
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-1/3, b=-12/5. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-41/15, 31/15, 4/5, 5/36)
Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-41/15, 31/15, 4/5, 5/36).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-ordered-four-results · candidate
13. Ordered Four Rational Operations
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=0/1, b=13/2. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(13/2, -13/2, 0/1, 0/1)
Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (13/2, -13/2, 0/1, 0/1).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-ordered-four-results · candidate
14. Ordered Four Rational Operations
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=1/2, b=-14/3. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-25/6, 31/6, -7/3, -3/28)
Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-25/6, 31/6, -7/3, -3/28).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-ordered-four-results · candidate
15. Ordered Four Rational Operations
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=2/3, b=15/4. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(53/12, -37/12, 5/2, 8/45)
Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (53/12, -37/12, 5/2, 8/45).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-ordered-four-results · candidate
16. Ordered Four Rational Operations
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=3/1, b=-16/5. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-1/5, 31/5, -48/5, -15/16)
Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-1/5, 31/5, -48/5, -15/16).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-ordered-four-results · candidate
17. Ordered Four Rational Operations
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=4/2, b=17/2. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(21/2, -13/2, 17/1, 4/17)
Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (21/2, -13/2, 17/1, 4/17).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-ordered-four-results · candidate
18. Ordered Four Rational Operations
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=5/3, b=-18/3. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-13/3, 23/3, -10/1, -5/18)
Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-13/3, 23/3, -10/1, -5/18).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-ordered-four-results · candidate
19. Ordered Four Rational Operations
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=6/1, b=19/4. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(43/4, 5/4, 57/2, 24/19)
Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (43/4, 5/4, 57/2, 24/19).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-ordered-four-results · candidate
20. Ordered Four Rational Operations
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=7/2, b=-20/5. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-1/2, 15/2, -14/1, -7/8)
Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-1/2, 15/2, -14/1, -7/8).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-ordered-four-results · candidate
21. Ordered Four Rational Operations
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=8/3, b=21/2. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(79/6, -47/6, 28/1, 16/63)
Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (79/6, -47/6, 28/1, 16/63).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-ordered-four-results · candidate
22. Ordered Four Rational Operations
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=9/1, b=-22/3. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(5/3, 49/3, -66/1, -27/22)
Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (5/3, 49/3, -66/1, -27/22).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-ordered-four-results · candidate
23. Ordered Four Rational Operations
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=10/2, b=23/4. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(43/4, -3/4, 115/4, 20/23)
Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (43/4, -3/4, 115/4, 20/23).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-ordered-four-results · candidate
24. Ordered Four Rational Operations
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=11/3, b=-24/5. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-17/15, 127/15, -88/5, -55/72)
Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (-17/15, 127/15, -88/5, -55/72).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-ordered-four-results · candidate
25. Ordered Four Rational Operations
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=12/1, b=25/2. Give (a+b, a−b, ab, a/b), using undefined for an undefined field. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(49/2, -1/2, 150/1, 24/25)
Perform exact addition, subtraction, multiplication, and guarded division in that order; division by zero is undefined. The exact result, retaining the requested order, is (49/2, -1/2, 150/1, 24/25).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-ordered-four-results · candidate
26. Two Explicit Parenthesis Orders
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=0/1, b=2/1, c=0/1. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(0/1, 0/1)
The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (0/1, 0/1).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-parentheses-pair · candidate
27. Two Explicit Parenthesis Orders
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=3/2, b=9/5, c=0/1. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(0/1, 3/2)
The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (0/1, 3/2).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-parentheses-pair · candidate
28. Two Explicit Parenthesis Orders
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=1/3, b=2/3, c=1/1. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/1, 1/1)
The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (1/1, 1/1).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-parentheses-pair · candidate
29. Two Explicit Parenthesis Orders
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=0/1, b=2/3, c=3/4. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(1/2, 1/2)
The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (1/2, 1/2).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-parentheses-pair · candidate
30. Two Explicit Parenthesis Orders
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-8/2, b=-3/3, c=5/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-25/2, -13/2)
The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-25/2, -13/2).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-parentheses-pair · candidate
31. Two Explicit Parenthesis Orders
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-7/2, b=-4/3, c=6/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-29/2, -15/2)
The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-29/2, -15/2).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-parentheses-pair · candidate
32. Two Explicit Parenthesis Orders
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-6/2, b=-5/3, c=7/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-49/3, -53/6)
The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-49/3, -53/6).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-parentheses-pair · candidate
33. Two Explicit Parenthesis Orders
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-5/2, b=-6/3, c=8/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-18/1, -21/2)
The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-18/1, -21/2).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-parentheses-pair · candidate
34. Two Explicit Parenthesis Orders
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-4/2, b=-7/3, c=9/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-39/2, -25/2)
The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-39/2, -25/2).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-parentheses-pair · candidate
35. Two Explicit Parenthesis Orders
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-3/2, b=-8/3, c=10/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-125/6, -89/6)
The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-125/6, -89/6).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-parentheses-pair · candidate
36. Two Explicit Parenthesis Orders
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-2/2, b=-9/3, c=11/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-22/1, -35/2)
The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-22/1, -35/2).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-parentheses-pair · candidate
37. Two Explicit Parenthesis Orders
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-1/2, b=-10/3, c=12/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-23/1, -41/2)
The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-23/1, -41/2).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-parentheses-pair · candidate
38. Two Explicit Parenthesis Orders
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=0/2, b=-11/3, c=13/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-143/6, -143/6)
The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-143/6, -143/6).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-parentheses-pair · candidate
39. Two Explicit Parenthesis Orders
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=1/2, b=-12/3, c=14/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-49/2, -55/2)
The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-49/2, -55/2).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-parentheses-pair · candidate
40. Two Explicit Parenthesis Orders
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=2/2, b=-13/3, c=15/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-25/1, -63/2)
The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-25/1, -63/2).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-parentheses-pair · candidate
41. Two Explicit Parenthesis Orders
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=3/2, b=-14/3, c=16/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-76/3, -215/6)
The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-76/3, -215/6).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-parentheses-pair · candidate
42. Two Explicit Parenthesis Orders
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=4/2, b=-15/3, c=17/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-51/2, -81/2)
The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-51/2, -81/2).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-parentheses-pair · candidate
43. Two Explicit Parenthesis Orders
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=5/2, b=-16/3, c=18/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-51/2, -91/2)
The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-51/2, -91/2).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-parentheses-pair · candidate
44. Two Explicit Parenthesis Orders
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=6/2, b=-17/3, c=19/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-76/3, -305/6)
The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-76/3, -305/6).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-parentheses-pair · candidate
45. Two Explicit Parenthesis Orders
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=7/2, b=-18/3, c=20/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-25/1, -113/2)
The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-25/1, -113/2).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-parentheses-pair · candidate
46. Two Explicit Parenthesis Orders
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=8/2, b=-19/3, c=21/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-49/2, -125/2)
The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-49/2, -125/2).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-parentheses-pair · candidate
47. Two Explicit Parenthesis Orders
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=9/2, b=-20/3, c=22/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-143/6, -413/6)
The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-143/6, -413/6).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-parentheses-pair · candidate
48. Two Explicit Parenthesis Orders
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=10/2, b=-21/3, c=23/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-23/1, -151/2)
The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-23/1, -151/2).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-parentheses-pair · candidate
49. Two Explicit Parenthesis Orders
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=11/2, b=-22/3, c=24/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-22/1, -165/2)
The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-22/1, -165/2).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-parentheses-pair · candidate
50. Two Explicit Parenthesis Orders
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=12/2, b=-23/3, c=25/2. Give ((a+b)c, a+bc), preserving parentheses. Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-125/6, -539/6)
The first expression adds before multiplying, while the second multiplies before adding; zero or unit factors do not change the declared evaluation order. The exact result, retaining the requested order, is (-125/6, -539/6).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-parentheses-pair · candidate
51. Inverse Candidate and Multiplicative Defect
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=0/1, candidate r=7/2. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(No, -1/1)
Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -1/1).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-inverse-candidate · candidate
52. Inverse Candidate and Multiplicative Defect
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=2/3, candidate r=0/1. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(No, -1/1)
Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -1/1).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-inverse-candidate · candidate
53. Inverse Candidate and Multiplicative Defect
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=3/2, candidate r=1/4. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(No, -5/8)
Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -5/8).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-inverse-candidate · candidate
54. Inverse Candidate and Multiplicative Defect
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-4/2, candidate r=-2/4. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, 0/1)
Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (Yes, 0/1).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-inverse-candidate · candidate
55. Inverse Candidate and Multiplicative Defect
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=5/2, candidate r=1/6. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(No, -7/12)
Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -7/12).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-inverse-candidate · candidate
56. Inverse Candidate and Multiplicative Defect
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-6/2, candidate r=1/7. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(No, -10/7)
Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -10/7).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-inverse-candidate · candidate
57. Inverse Candidate and Multiplicative Defect
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=7/2, candidate r=2/7. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, 0/1)
Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (Yes, 0/1).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-inverse-candidate · candidate
58. Inverse Candidate and Multiplicative Defect
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-8/2, candidate r=1/9. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(No, -13/9)
Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -13/9).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-inverse-candidate · candidate
59. Inverse Candidate and Multiplicative Defect
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=9/2, candidate r=1/10. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(No, -11/20)
Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -11/20).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-inverse-candidate · candidate
60. Inverse Candidate and Multiplicative Defect
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-10/2, candidate r=-2/10. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, 0/1)
Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (Yes, 0/1).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-inverse-candidate · candidate
61. Inverse Candidate and Multiplicative Defect
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=11/2, candidate r=1/12. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(No, -13/24)
Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -13/24).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-inverse-candidate · candidate
62. Inverse Candidate and Multiplicative Defect
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-12/2, candidate r=1/13. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(No, -19/13)
Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -19/13).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-inverse-candidate · candidate
63. Inverse Candidate and Multiplicative Defect
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=13/2, candidate r=2/13. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, 0/1)
Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (Yes, 0/1).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-inverse-candidate · candidate
64. Inverse Candidate and Multiplicative Defect
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-14/2, candidate r=1/15. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(No, -22/15)
Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -22/15).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-inverse-candidate · candidate
65. Inverse Candidate and Multiplicative Defect
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=15/2, candidate r=1/16. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(No, -17/32)
Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -17/32).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-inverse-candidate · candidate
66. Inverse Candidate and Multiplicative Defect
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-16/2, candidate r=-2/16. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, 0/1)
Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (Yes, 0/1).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-inverse-candidate · candidate
67. Inverse Candidate and Multiplicative Defect
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=17/2, candidate r=1/18. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(No, -19/36)
Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -19/36).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-inverse-candidate · candidate
68. Inverse Candidate and Multiplicative Defect
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-18/2, candidate r=1/19. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(No, -28/19)
Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -28/19).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-inverse-candidate · candidate
69. Inverse Candidate and Multiplicative Defect
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=19/2, candidate r=2/19. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, 0/1)
Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (Yes, 0/1).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-inverse-candidate · candidate
70. Inverse Candidate and Multiplicative Defect
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-20/2, candidate r=1/21. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(No, -31/21)
Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -31/21).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-inverse-candidate · candidate
71. Inverse Candidate and Multiplicative Defect
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=21/2, candidate r=1/22. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(No, -23/44)
Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -23/44).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-inverse-candidate · candidate
72. Inverse Candidate and Multiplicative Defect
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-22/2, candidate r=-2/22. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, 0/1)
Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (Yes, 0/1).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-inverse-candidate · candidate
73. Inverse Candidate and Multiplicative Defect
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=23/2, candidate r=1/24. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(No, -25/48)
Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -25/48).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-inverse-candidate · candidate
74. Inverse Candidate and Multiplicative Defect
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=-24/2, candidate r=1/25. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(No, -37/25)
Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (No, -37/25).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-inverse-candidate · candidate
75. Inverse Candidate and Multiplicative Defect
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. a=25/2, candidate r=2/25. Give (is r a multiplicative inverse of a?, ar−1). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, 0/1)
Zero has no multiplicative inverse; a nonzero operand still requires the product to equal exactly one. Keep the defect’s sign. The exact result, retaining the requested order, is (Yes, 0/1).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-inverse-candidate · candidate
76. Q Results and Natural-System Eligibility
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=0, b=0. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
View answer and explanation
(0, Yes, undefined, undefined)
A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (0, Yes, undefined, undefined).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-system-membership · candidate
77. Q Results and Natural-System Eligibility
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=1, b=0. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
View answer and explanation
(1, Yes, undefined, undefined)
A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (1, Yes, undefined, undefined).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-system-membership · candidate
78. Q Results and Natural-System Eligibility
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=1, b=2. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
View answer and explanation
(-1, No, 1/2, No)
A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (-1, No, 1/2, No).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-system-membership · candidate
79. Q Results and Natural-System Eligibility
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=6, b=3. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
View answer and explanation
(3, Yes, 2/1, Yes)
A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (3, Yes, 2/1, Yes).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-system-membership · candidate
80. Q Results and Natural-System Eligibility
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=0, b=3. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
View answer and explanation
(-3, No, 0/1, Yes)
A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (-3, No, 0/1, Yes).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-system-membership · candidate
81. Q Results and Natural-System Eligibility
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=2, b=1. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
View answer and explanation
(1, Yes, 2/1, Yes)
A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (1, Yes, 2/1, Yes).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-system-membership · candidate
82. Q Results and Natural-System Eligibility
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=3, b=2. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
View answer and explanation
(1, Yes, 3/2, No)
A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (1, Yes, 3/2, No).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-system-membership · candidate
83. Q Results and Natural-System Eligibility
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=4, b=3. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
View answer and explanation
(1, Yes, 4/3, No)
A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (1, Yes, 4/3, No).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-system-membership · candidate
84. Q Results and Natural-System Eligibility
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=5, b=4. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
View answer and explanation
(1, Yes, 5/4, No)
A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (1, Yes, 5/4, No).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-system-membership · candidate
85. Q Results and Natural-System Eligibility
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=6, b=5. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
View answer and explanation
(1, Yes, 6/5, No)
A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (1, Yes, 6/5, No).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-system-membership · candidate
86. Q Results and Natural-System Eligibility
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=7, b=1. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
View answer and explanation
(6, Yes, 7/1, Yes)
A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (6, Yes, 7/1, Yes).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-system-membership · candidate
87. Q Results and Natural-System Eligibility
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=8, b=2. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
View answer and explanation
(6, Yes, 4/1, Yes)
A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (6, Yes, 4/1, Yes).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-system-membership · candidate
88. Q Results and Natural-System Eligibility
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=9, b=3. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
View answer and explanation
(6, Yes, 3/1, Yes)
A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (6, Yes, 3/1, Yes).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-system-membership · candidate
89. Q Results and Natural-System Eligibility
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=10, b=4. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
View answer and explanation
(6, Yes, 5/2, No)
A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (6, Yes, 5/2, No).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-system-membership · candidate
90. Q Results and Natural-System Eligibility
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=11, b=5. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
View answer and explanation
(6, Yes, 11/5, No)
A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (6, Yes, 11/5, No).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-system-membership · candidate
91. Q Results and Natural-System Eligibility
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=12, b=1. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
View answer and explanation
(11, Yes, 12/1, Yes)
A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (11, Yes, 12/1, Yes).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-system-membership · candidate
92. Q Results and Natural-System Eligibility
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=13, b=2. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
View answer and explanation
(11, Yes, 13/2, No)
A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (11, Yes, 13/2, No).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-system-membership · candidate
93. Q Results and Natural-System Eligibility
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=14, b=3. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
View answer and explanation
(11, Yes, 14/3, No)
A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (11, Yes, 14/3, No).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-system-membership · candidate
94. Q Results and Natural-System Eligibility
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=15, b=4. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
View answer and explanation
(11, Yes, 15/4, No)
A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (11, Yes, 15/4, No).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-system-membership · candidate
95. Q Results and Natural-System Eligibility
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=16, b=5. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
View answer and explanation
(11, Yes, 16/5, No)
A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (11, Yes, 16/5, No).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-system-membership · candidate
96. Q Results and Natural-System Eligibility
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=17, b=1. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
View answer and explanation
(16, Yes, 17/1, Yes)
A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (16, Yes, 17/1, Yes).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-system-membership · candidate
97. Q Results and Natural-System Eligibility
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=18, b=2. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
View answer and explanation
(16, Yes, 9/1, Yes)
A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (16, Yes, 9/1, Yes).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-system-membership · candidate
98. Q Results and Natural-System Eligibility
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=19, b=3. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
View answer and explanation
(16, Yes, 19/3, No)
A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (16, Yes, 19/3, No).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-system-membership · candidate
99. Q Results and Natural-System Eligibility
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=20, b=4. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
View answer and explanation
(16, Yes, 5/1, Yes)
A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (16, Yes, 5/1, Yes).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-system-membership · candidate
100. Q Results and Natural-System Eligibility
All operations and evaluations are exact in Q, not truncated natural-number arithmetic. N includes 0. Natural inputs a=21, b=5. Give (a−b, does the difference belong to N?, a/b, does the quotient belong to N?). Use undefined for an undefined quotient or its eligibility.
View answer and explanation
(16, Yes, 21/5, No)
A negative difference is outside N and a noninteger quotient is outside N; neither truncated subtraction nor integer division is used, and division by zero stays undefined. The exact result, retaining the requested order, is (16, Yes, 21/5, No).
Reviewed step-by-step derivation
Arithmetic operations · qf-density11-arithmetic-system-membership · candidate
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