# Exact Rational Operations and Natural Eligibility

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r02-arithmetic-operations-0100/)

Arithmetic operations · qf-density11-arithmetic-system-membership · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261011-r02-arithmetic-operations-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261008-advance-20261011-r02-arithmetic-operations-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261011-r02-arithmetic-operations-foundations/validation-report.json)