# Euclidean Division and Signed Integer Protocols

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-20261009-advance-20261013-r02-integer-foundations · 2026-10-09 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Euclidean Integer Quotient and Remainder

Integer a=0, positive b=5. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.

View answer and explanation(0, 0)

If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (0, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0001/)

Integer · qf-density13-integer-euclidean-divmod · candidate





## 2. Euclidean Integer Quotient and Remainder

Integer a=-7, positive b=3. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.

View answer and explanation(-3, 2)

If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (-3, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0002/)

Integer · qf-density13-integer-euclidean-divmod · candidate





## 3. Euclidean Integer Quotient and Remainder

Integer a=3, positive b=4. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.

View answer and explanation(0, 3)

If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (0, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0003/)

Integer · qf-density13-integer-euclidean-divmod · candidate





## 4. Euclidean Integer Quotient and Remainder

Integer a=-4, positive b=5. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.

View answer and explanation(-1, 1)

If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (-1, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0004/)

Integer · qf-density13-integer-euclidean-divmod · candidate





## 5. Euclidean Integer Quotient and Remainder

Integer a=5, positive b=6. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.

View answer and explanation(0, 5)

If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (0, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0005/)

Integer · qf-density13-integer-euclidean-divmod · candidate





## 6. Euclidean Integer Quotient and Remainder

Integer a=-6, positive b=2. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.

View answer and explanation(-3, 0)

If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (-3, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0006/)

Integer · qf-density13-integer-euclidean-divmod · candidate





## 7. Euclidean Integer Quotient and Remainder

Integer a=7, positive b=3. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.

View answer and explanation(2, 1)

If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (2, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0007/)

Integer · qf-density13-integer-euclidean-divmod · candidate





## 8. Euclidean Integer Quotient and Remainder

Integer a=-8, positive b=4. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.

View answer and explanation(-2, 0)

If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (-2, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0008/)

Integer · qf-density13-integer-euclidean-divmod · candidate





## 9. Euclidean Integer Quotient and Remainder

Integer a=9, positive b=5. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.

View answer and explanation(1, 4)

If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (1, 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0009/)

Integer · qf-density13-integer-euclidean-divmod · candidate





## 10. Euclidean Integer Quotient and Remainder

Integer a=-10, positive b=6. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.

View answer and explanation(-2, 2)

If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (-2, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0010/)

Integer · qf-density13-integer-euclidean-divmod · candidate





## 11. Euclidean Integer Quotient and Remainder

Integer a=11, positive b=2. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.

View answer and explanation(5, 1)

If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (5, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0011/)

Integer · qf-density13-integer-euclidean-divmod · candidate





## 12. Euclidean Integer Quotient and Remainder

Integer a=-12, positive b=3. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.

View answer and explanation(-4, 0)

If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (-4, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0012/)

Integer · qf-density13-integer-euclidean-divmod · candidate





## 13. Euclidean Integer Quotient and Remainder

Integer a=13, positive b=4. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.

View answer and explanation(3, 1)

If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (3, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0013/)

Integer · qf-density13-integer-euclidean-divmod · candidate





## 14. Euclidean Integer Quotient and Remainder

Integer a=-14, positive b=5. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.

View answer and explanation(-3, 1)

If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (-3, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0014/)

Integer · qf-density13-integer-euclidean-divmod · candidate





## 15. Euclidean Integer Quotient and Remainder

Integer a=15, positive b=6. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.

View answer and explanation(2, 3)

If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (2, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0015/)

Integer · qf-density13-integer-euclidean-divmod · candidate





## 16. Euclidean Integer Quotient and Remainder

Integer a=-16, positive b=2. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.

View answer and explanation(-8, 0)

If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (-8, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0016/)

Integer · qf-density13-integer-euclidean-divmod · candidate





## 17. Euclidean Integer Quotient and Remainder

Integer a=17, positive b=3. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.

View answer and explanation(5, 2)

If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (5, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0017/)

Integer · qf-density13-integer-euclidean-divmod · candidate





## 18. Euclidean Integer Quotient and Remainder

Integer a=-18, positive b=4. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.

View answer and explanation(-5, 2)

If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (-5, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0018/)

Integer · qf-density13-integer-euclidean-divmod · candidate





## 19. Euclidean Integer Quotient and Remainder

Integer a=19, positive b=5. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.

View answer and explanation(3, 4)

If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (3, 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0019/)

Integer · qf-density13-integer-euclidean-divmod · candidate





## 20. Euclidean Integer Quotient and Remainder

Integer a=-20, positive b=6. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.

View answer and explanation(-4, 4)

If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (-4, 4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0020/)

Integer · qf-density13-integer-euclidean-divmod · candidate





## 21. Euclidean Integer Quotient and Remainder

Integer a=21, positive b=2. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.

View answer and explanation(10, 1)

If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (10, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0021/)

Integer · qf-density13-integer-euclidean-divmod · candidate





## 22. Euclidean Integer Quotient and Remainder

Integer a=-22, positive b=3. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.

View answer and explanation(-8, 2)

If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (-8, 2).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0022/)

Integer · qf-density13-integer-euclidean-divmod · candidate





## 23. Euclidean Integer Quotient and Remainder

Integer a=23, positive b=4. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.

View answer and explanation(5, 3)

If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (5, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0023/)

Integer · qf-density13-integer-euclidean-divmod · candidate





## 24. Euclidean Integer Quotient and Remainder

Integer a=-24, positive b=5. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.

View answer and explanation(-5, 1)

If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (-5, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0024/)

Integer · qf-density13-integer-euclidean-divmod · candidate





## 25. Euclidean Integer Quotient and Remainder

Integer a=25, positive b=6. Give (q,r) with a=bq+r and 0≤r<b, not a negative remainder from truncation toward zero.

View answer and explanation(4, 1)

If a truncated division gives a negative remainder, adjust it into 0,…,b−1 and correct the quotient. Finite arithmetic is not an integer-ring axiom proof. The exact result, retaining the requested order, is (4, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0025/)

Integer · qf-density13-integer-euclidean-divmod · candidate





## 26. Endpoint, Net Displacement, and Step Magnitude

Integer start=0, signed steps s1=0, s2=0. Give (endpoint, s1+s2, |s1|+|s2|).

View answer and explanation(0, 0, 0)

Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (0, 0, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0026/)

Integer · qf-density13-integer-two-step-displacement · candidate





## 27. Endpoint, Net Displacement, and Step Magnitude

Integer start=1, signed steps s1=3, s2=-3. Give (endpoint, s1+s2, |s1|+|s2|).

View answer and explanation(1, 0, 6)

Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (1, 0, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0027/)

Integer · qf-density13-integer-two-step-displacement · candidate





## 28. Endpoint, Net Displacement, and Step Magnitude

Integer start=-10, signed steps s1=3, s2=-3. Give (endpoint, s1+s2, |s1|+|s2|).

View answer and explanation(-10, 0, 6)

Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (-10, 0, 6).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0028/)

Integer · qf-density13-integer-two-step-displacement · candidate





## 29. Endpoint, Net Displacement, and Step Magnitude

Integer start=-9, signed steps s1=4, s2=-4. Give (endpoint, s1+s2, |s1|+|s2|).

View answer and explanation(-9, 0, 8)

Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (-9, 0, 8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0029/)

Integer · qf-density13-integer-two-step-displacement · candidate





## 30. Endpoint, Net Displacement, and Step Magnitude

Integer start=-8, signed steps s1=5, s2=-5. Give (endpoint, s1+s2, |s1|+|s2|).

View answer and explanation(-8, 0, 10)

Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (-8, 0, 10).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0030/)

Integer · qf-density13-integer-two-step-displacement · candidate





## 31. Endpoint, Net Displacement, and Step Magnitude

Integer start=-7, signed steps s1=6, s2=-1. Give (endpoint, s1+s2, |s1|+|s2|).

View answer and explanation(-2, 5, 7)

Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (-2, 5, 7).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0031/)

Integer · qf-density13-integer-two-step-displacement · candidate





## 32. Endpoint, Net Displacement, and Step Magnitude

Integer start=-6, signed steps s1=7, s2=-2. Give (endpoint, s1+s2, |s1|+|s2|).

View answer and explanation(-1, 5, 9)

Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (-1, 5, 9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0032/)

Integer · qf-density13-integer-two-step-displacement · candidate





## 33. Endpoint, Net Displacement, and Step Magnitude

Integer start=-5, signed steps s1=8, s2=-3. Give (endpoint, s1+s2, |s1|+|s2|).

View answer and explanation(0, 5, 11)

Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (0, 5, 11).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0033/)

Integer · qf-density13-integer-two-step-displacement · candidate





## 34. Endpoint, Net Displacement, and Step Magnitude

Integer start=-4, signed steps s1=9, s2=-4. Give (endpoint, s1+s2, |s1|+|s2|).

View answer and explanation(1, 5, 13)

Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (1, 5, 13).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0034/)

Integer · qf-density13-integer-two-step-displacement · candidate





## 35. Endpoint, Net Displacement, and Step Magnitude

Integer start=-3, signed steps s1=10, s2=-5. Give (endpoint, s1+s2, |s1|+|s2|).

View answer and explanation(2, 5, 15)

Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (2, 5, 15).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0035/)

Integer · qf-density13-integer-two-step-displacement · candidate





## 36. Endpoint, Net Displacement, and Step Magnitude

Integer start=-2, signed steps s1=11, s2=-1. Give (endpoint, s1+s2, |s1|+|s2|).

View answer and explanation(8, 10, 12)

Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (8, 10, 12).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0036/)

Integer · qf-density13-integer-two-step-displacement · candidate





## 37. Endpoint, Net Displacement, and Step Magnitude

Integer start=-1, signed steps s1=12, s2=-2. Give (endpoint, s1+s2, |s1|+|s2|).

View answer and explanation(9, 10, 14)

Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (9, 10, 14).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0037/)

Integer · qf-density13-integer-two-step-displacement · candidate





## 38. Endpoint, Net Displacement, and Step Magnitude

Integer start=0, signed steps s1=13, s2=-3. Give (endpoint, s1+s2, |s1|+|s2|).

View answer and explanation(10, 10, 16)

Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (10, 10, 16).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0038/)

Integer · qf-density13-integer-two-step-displacement · candidate





## 39. Endpoint, Net Displacement, and Step Magnitude

Integer start=1, signed steps s1=14, s2=-4. Give (endpoint, s1+s2, |s1|+|s2|).

View answer and explanation(11, 10, 18)

Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (11, 10, 18).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0039/)

Integer · qf-density13-integer-two-step-displacement · candidate





## 40. Endpoint, Net Displacement, and Step Magnitude

Integer start=2, signed steps s1=15, s2=-5. Give (endpoint, s1+s2, |s1|+|s2|).

View answer and explanation(12, 10, 20)

Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (12, 10, 20).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0040/)

Integer · qf-density13-integer-two-step-displacement · candidate





## 41. Endpoint, Net Displacement, and Step Magnitude

Integer start=3, signed steps s1=16, s2=-1. Give (endpoint, s1+s2, |s1|+|s2|).

View answer and explanation(18, 15, 17)

Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (18, 15, 17).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0041/)

Integer · qf-density13-integer-two-step-displacement · candidate





## 42. Endpoint, Net Displacement, and Step Magnitude

Integer start=4, signed steps s1=17, s2=-2. Give (endpoint, s1+s2, |s1|+|s2|).

View answer and explanation(19, 15, 19)

Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (19, 15, 19).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0042/)

Integer · qf-density13-integer-two-step-displacement · candidate





## 43. Endpoint, Net Displacement, and Step Magnitude

Integer start=5, signed steps s1=18, s2=-3. Give (endpoint, s1+s2, |s1|+|s2|).

View answer and explanation(20, 15, 21)

Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (20, 15, 21).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0043/)

Integer · qf-density13-integer-two-step-displacement · candidate





## 44. Endpoint, Net Displacement, and Step Magnitude

Integer start=6, signed steps s1=19, s2=-4. Give (endpoint, s1+s2, |s1|+|s2|).

View answer and explanation(21, 15, 23)

Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (21, 15, 23).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0044/)

Integer · qf-density13-integer-two-step-displacement · candidate





## 45. Endpoint, Net Displacement, and Step Magnitude

Integer start=7, signed steps s1=20, s2=-5. Give (endpoint, s1+s2, |s1|+|s2|).

View answer and explanation(22, 15, 25)

Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (22, 15, 25).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0045/)

Integer · qf-density13-integer-two-step-displacement · candidate





## 46. Endpoint, Net Displacement, and Step Magnitude

Integer start=8, signed steps s1=21, s2=-1. Give (endpoint, s1+s2, |s1|+|s2|).

View answer and explanation(28, 20, 22)

Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (28, 20, 22).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0046/)

Integer · qf-density13-integer-two-step-displacement · candidate





## 47. Endpoint, Net Displacement, and Step Magnitude

Integer start=9, signed steps s1=22, s2=-2. Give (endpoint, s1+s2, |s1|+|s2|).

View answer and explanation(29, 20, 24)

Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (29, 20, 24).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0047/)

Integer · qf-density13-integer-two-step-displacement · candidate





## 48. Endpoint, Net Displacement, and Step Magnitude

Integer start=10, signed steps s1=23, s2=-3. Give (endpoint, s1+s2, |s1|+|s2|).

View answer and explanation(30, 20, 26)

Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (30, 20, 26).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0048/)

Integer · qf-density13-integer-two-step-displacement · candidate





## 49. Endpoint, Net Displacement, and Step Magnitude

Integer start=11, signed steps s1=24, s2=-4. Give (endpoint, s1+s2, |s1|+|s2|).

View answer and explanation(31, 20, 28)

Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (31, 20, 28).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0049/)

Integer · qf-density13-integer-two-step-displacement · candidate





## 50. Endpoint, Net Displacement, and Step Magnitude

Integer start=12, signed steps s1=25, s2=-5. Give (endpoint, s1+s2, |s1|+|s2|).

View answer and explanation(32, 20, 30)

Add signed steps for displacement and endpoint, but add their absolute values for step magnitude; returning to the start can still have positive step magnitude. The exact result, retaining the requested order, is (32, 20, 30).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0050/)

Integer · qf-density13-integer-two-step-displacement · candidate





## 51. Ordered Opposites and Two Difference Directions

Integers a=0, b=0. Give (−a, −b, a−b, b−a), retaining signs and direction.

View answer and explanation(0, 0, 0, 0)

Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (0, 0, 0, 0).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0051/)

Integer · qf-density13-integer-opposite-difference · candidate





## 52. Ordered Opposites and Two Difference Directions

Integers a=-11, b=0. Give (−a, −b, a−b, b−a), retaining signs and direction.

View answer and explanation(11, 0, -11, 11)

Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (11, 0, -11, 11).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0052/)

Integer · qf-density13-integer-opposite-difference · candidate





## 53. Ordered Opposites and Two Difference Directions

Integers a=-10, b=-1. Give (−a, −b, a−b, b−a), retaining signs and direction.

View answer and explanation(10, 1, -9, 9)

Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (10, 1, -9, 9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0053/)

Integer · qf-density13-integer-opposite-difference · candidate





## 54. Ordered Opposites and Two Difference Directions

Integers a=-9, b=-2. Give (−a, −b, a−b, b−a), retaining signs and direction.

View answer and explanation(9, 2, -7, 7)

Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (9, 2, -7, 7).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0054/)

Integer · qf-density13-integer-opposite-difference · candidate





## 55. Ordered Opposites and Two Difference Directions

Integers a=-8, b=-3. Give (−a, −b, a−b, b−a), retaining signs and direction.

View answer and explanation(8, 3, -5, 5)

Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (8, 3, -5, 5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0055/)

Integer · qf-density13-integer-opposite-difference · candidate





## 56. Ordered Opposites and Two Difference Directions

Integers a=-7, b=-4. Give (−a, −b, a−b, b−a), retaining signs and direction.

View answer and explanation(7, 4, -3, 3)

Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (7, 4, -3, 3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0056/)

Integer · qf-density13-integer-opposite-difference · candidate





## 57. Ordered Opposites and Two Difference Directions

Integers a=-6, b=-5. Give (−a, −b, a−b, b−a), retaining signs and direction.

View answer and explanation(6, 5, -1, 1)

Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (6, 5, -1, 1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0057/)

Integer · qf-density13-integer-opposite-difference · candidate





## 58. Ordered Opposites and Two Difference Directions

Integers a=-5, b=-6. Give (−a, −b, a−b, b−a), retaining signs and direction.

View answer and explanation(5, 6, 1, -1)

Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (5, 6, 1, -1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0058/)

Integer · qf-density13-integer-opposite-difference · candidate





## 59. Ordered Opposites and Two Difference Directions

Integers a=-4, b=-7. Give (−a, −b, a−b, b−a), retaining signs and direction.

View answer and explanation(4, 7, 3, -3)

Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (4, 7, 3, -3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0059/)

Integer · qf-density13-integer-opposite-difference · candidate





## 60. Ordered Opposites and Two Difference Directions

Integers a=-3, b=-8. Give (−a, −b, a−b, b−a), retaining signs and direction.

View answer and explanation(3, 8, 5, -5)

Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (3, 8, 5, -5).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0060/)

Integer · qf-density13-integer-opposite-difference · candidate





## 61. Ordered Opposites and Two Difference Directions

Integers a=-2, b=-9. Give (−a, −b, a−b, b−a), retaining signs and direction.

View answer and explanation(2, 9, 7, -7)

Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (2, 9, 7, -7).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0061/)

Integer · qf-density13-integer-opposite-difference · candidate





## 62. Ordered Opposites and Two Difference Directions

Integers a=-1, b=-10. Give (−a, −b, a−b, b−a), retaining signs and direction.

View answer and explanation(1, 10, 9, -9)

Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (1, 10, 9, -9).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0062/)

Integer · qf-density13-integer-opposite-difference · candidate





## 63. Ordered Opposites and Two Difference Directions

Integers a=0, b=-11. Give (−a, −b, a−b, b−a), retaining signs and direction.

View answer and explanation(0, 11, 11, -11)

Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (0, 11, 11, -11).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0063/)

Integer · qf-density13-integer-opposite-difference · candidate





## 64. Ordered Opposites and Two Difference Directions

Integers a=1, b=-12. Give (−a, −b, a−b, b−a), retaining signs and direction.

View answer and explanation(-1, 12, 13, -13)

Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (-1, 12, 13, -13).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0064/)

Integer · qf-density13-integer-opposite-difference · candidate





## 65. Ordered Opposites and Two Difference Directions

Integers a=2, b=-13. Give (−a, −b, a−b, b−a), retaining signs and direction.

View answer and explanation(-2, 13, 15, -15)

Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (-2, 13, 15, -15).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0065/)

Integer · qf-density13-integer-opposite-difference · candidate





## 66. Ordered Opposites and Two Difference Directions

Integers a=3, b=-14. Give (−a, −b, a−b, b−a), retaining signs and direction.

View answer and explanation(-3, 14, 17, -17)

Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (-3, 14, 17, -17).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0066/)

Integer · qf-density13-integer-opposite-difference · candidate





## 67. Ordered Opposites and Two Difference Directions

Integers a=4, b=-15. Give (−a, −b, a−b, b−a), retaining signs and direction.

View answer and explanation(-4, 15, 19, -19)

Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (-4, 15, 19, -19).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0067/)

Integer · qf-density13-integer-opposite-difference · candidate





## 68. Ordered Opposites and Two Difference Directions

Integers a=5, b=-16. Give (−a, −b, a−b, b−a), retaining signs and direction.

View answer and explanation(-5, 16, 21, -21)

Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (-5, 16, 21, -21).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0068/)

Integer · qf-density13-integer-opposite-difference · candidate





## 69. Ordered Opposites and Two Difference Directions

Integers a=6, b=-17. Give (−a, −b, a−b, b−a), retaining signs and direction.

View answer and explanation(-6, 17, 23, -23)

Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (-6, 17, 23, -23).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0069/)

Integer · qf-density13-integer-opposite-difference · candidate





## 70. Ordered Opposites and Two Difference Directions

Integers a=7, b=-18. Give (−a, −b, a−b, b−a), retaining signs and direction.

View answer and explanation(-7, 18, 25, -25)

Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (-7, 18, 25, -25).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0070/)

Integer · qf-density13-integer-opposite-difference · candidate





## 71. Ordered Opposites and Two Difference Directions

Integers a=8, b=-19. Give (−a, −b, a−b, b−a), retaining signs and direction.

View answer and explanation(-8, 19, 27, -27)

Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (-8, 19, 27, -27).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0071/)

Integer · qf-density13-integer-opposite-difference · candidate





## 72. Ordered Opposites and Two Difference Directions

Integers a=9, b=-20. Give (−a, −b, a−b, b−a), retaining signs and direction.

View answer and explanation(-9, 20, 29, -29)

Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (-9, 20, 29, -29).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0072/)

Integer · qf-density13-integer-opposite-difference · candidate





## 73. Ordered Opposites and Two Difference Directions

Integers a=10, b=-21. Give (−a, −b, a−b, b−a), retaining signs and direction.

View answer and explanation(-10, 21, 31, -31)

Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (-10, 21, 31, -31).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0073/)

Integer · qf-density13-integer-opposite-difference · candidate





## 74. Ordered Opposites and Two Difference Directions

Integers a=11, b=-22. Give (−a, −b, a−b, b−a), retaining signs and direction.

View answer and explanation(-11, 22, 33, -33)

Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (-11, 22, 33, -33).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0074/)

Integer · qf-density13-integer-opposite-difference · candidate





## 75. Ordered Opposites and Two Difference Directions

Integers a=12, b=-23. Give (−a, −b, a−b, b−a), retaining signs and direction.

View answer and explanation(-12, 23, 35, -35)

Negate each operand separately and subtract in the requested order; this finite computation does not prove all integer algebra laws. The exact result, retaining the requested order, is (-12, 23, 35, -35).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0075/)

Integer · qf-density13-integer-opposite-difference · candidate





## 76. Rational Candidates Split by Integer Eligibility

Ordered representations=[0/1, -2/1, 1/2, 1/-3] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.

View answer and explanation([Yes, Yes, No, No], [0, -2], [1/2, -1/3])

Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [0, -2], [1/2, -1/3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0076/)

Integer · qf-density13-integer-rational-grid-split · candidate





## 77. Rational Candidates Split by Integer Eligibility

Ordered representations=[2/-2, 3/1, 4/-2, 5/2] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.

View answer and explanation([Yes, Yes, Yes, No], [-1, 3, -2], [5/2])

Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, Yes, No], [-1, 3, -2], [5/2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0077/)

Integer · qf-density13-integer-rational-grid-split · candidate





## 78. Rational Candidates Split by Integer Eligibility

Ordered representations=[-10/1, 4/-2, 3/7, -4/-9] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.

View answer and explanation([Yes, Yes, No, No], [-10, -2], [3/7, 4/9])

Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [-10, -2], [3/7, 4/9]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0078/)

Integer · qf-density13-integer-rational-grid-split · candidate





## 79. Rational Candidates Split by Integer Eligibility

Ordered representations=[-9/1, 6/-2, 4/9, -5/-11] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.

View answer and explanation([Yes, Yes, No, No], [-9, -3], [4/9, 5/11])

Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [-9, -3], [4/9, 5/11]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0079/)

Integer · qf-density13-integer-rational-grid-split · candidate





## 80. Rational Candidates Split by Integer Eligibility

Ordered representations=[-8/1, 8/-2, 5/11, -6/-13] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.

View answer and explanation([Yes, Yes, No, No], [-8, -4], [5/11, 6/13])

Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [-8, -4], [5/11, 6/13]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0080/)

Integer · qf-density13-integer-rational-grid-split · candidate





## 81. Rational Candidates Split by Integer Eligibility

Ordered representations=[-7/1, 10/-2, 6/13, -7/-15] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.

View answer and explanation([Yes, Yes, No, No], [-7, -5], [6/13, 7/15])

Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [-7, -5], [6/13, 7/15]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0081/)

Integer · qf-density13-integer-rational-grid-split · candidate





## 82. Rational Candidates Split by Integer Eligibility

Ordered representations=[-6/1, 12/-2, 7/15, -8/-17] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.

View answer and explanation([Yes, Yes, No, No], [-6, -6], [7/15, 8/17])

Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [-6, -6], [7/15, 8/17]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0082/)

Integer · qf-density13-integer-rational-grid-split · candidate





## 83. Rational Candidates Split by Integer Eligibility

Ordered representations=[-5/1, 14/-2, 8/17, -9/-19] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.

View answer and explanation([Yes, Yes, No, No], [-5, -7], [8/17, 9/19])

Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [-5, -7], [8/17, 9/19]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0083/)

Integer · qf-density13-integer-rational-grid-split · candidate





## 84. Rational Candidates Split by Integer Eligibility

Ordered representations=[-4/1, 16/-2, 9/19, -10/-21] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.

View answer and explanation([Yes, Yes, No, No], [-4, -8], [9/19, 10/21])

Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [-4, -8], [9/19, 10/21]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0084/)

Integer · qf-density13-integer-rational-grid-split · candidate





## 85. Rational Candidates Split by Integer Eligibility

Ordered representations=[-3/1, 18/-2, 10/21, -11/-23] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.

View answer and explanation([Yes, Yes, No, No], [-3, -9], [10/21, 11/23])

Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [-3, -9], [10/21, 11/23]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0085/)

Integer · qf-density13-integer-rational-grid-split · candidate





## 86. Rational Candidates Split by Integer Eligibility

Ordered representations=[-2/1, 20/-2, 11/23, -12/-25] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.

View answer and explanation([Yes, Yes, No, No], [-2, -10], [11/23, 12/25])

Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [-2, -10], [11/23, 12/25]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0086/)

Integer · qf-density13-integer-rational-grid-split · candidate





## 87. Rational Candidates Split by Integer Eligibility

Ordered representations=[-1/1, 22/-2, 12/25, -13/-27] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.

View answer and explanation([Yes, Yes, No, No], [-1, -11], [12/25, 13/27])

Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [-1, -11], [12/25, 13/27]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0087/)

Integer · qf-density13-integer-rational-grid-split · candidate





## 88. Rational Candidates Split by Integer Eligibility

Ordered representations=[0/1, 24/-2, 13/27, -14/-29] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.

View answer and explanation([Yes, Yes, No, No], [0, -12], [13/27, 14/29])

Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [0, -12], [13/27, 14/29]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0088/)

Integer · qf-density13-integer-rational-grid-split · candidate





## 89. Rational Candidates Split by Integer Eligibility

Ordered representations=[1/1, 26/-2, 14/29, -15/-31] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.

View answer and explanation([Yes, Yes, No, No], [1, -13], [14/29, 15/31])

Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [1, -13], [14/29, 15/31]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0089/)

Integer · qf-density13-integer-rational-grid-split · candidate





## 90. Rational Candidates Split by Integer Eligibility

Ordered representations=[2/1, 28/-2, 15/31, -16/-33] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.

View answer and explanation([Yes, Yes, No, No], [2, -14], [15/31, 16/33])

Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [2, -14], [15/31, 16/33]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0090/)

Integer · qf-density13-integer-rational-grid-split · candidate





## 91. Rational Candidates Split by Integer Eligibility

Ordered representations=[3/1, 30/-2, 16/33, -17/-35] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.

View answer and explanation([Yes, Yes, No, No], [3, -15], [16/33, 17/35])

Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [3, -15], [16/33, 17/35]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0091/)

Integer · qf-density13-integer-rational-grid-split · candidate





## 92. Rational Candidates Split by Integer Eligibility

Ordered representations=[4/1, 32/-2, 17/35, -18/-37] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.

View answer and explanation([Yes, Yes, No, No], [4, -16], [17/35, 18/37])

Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [4, -16], [17/35, 18/37]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0092/)

Integer · qf-density13-integer-rational-grid-split · candidate





## 93. Rational Candidates Split by Integer Eligibility

Ordered representations=[5/1, 34/-2, 18/37, -19/-39] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.

View answer and explanation([Yes, Yes, No, No], [5, -17], [18/37, 19/39])

Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [5, -17], [18/37, 19/39]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0093/)

Integer · qf-density13-integer-rational-grid-split · candidate





## 94. Rational Candidates Split by Integer Eligibility

Ordered representations=[6/1, 36/-2, 19/39, -20/-41] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.

View answer and explanation([Yes, Yes, No, No], [6, -18], [19/39, 20/41])

Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [6, -18], [19/39, 20/41]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0094/)

Integer · qf-density13-integer-rational-grid-split · candidate





## 95. Rational Candidates Split by Integer Eligibility

Ordered representations=[7/1, 38/-2, 20/41, -21/-43] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.

View answer and explanation([Yes, Yes, No, No], [7, -19], [20/41, 21/43])

Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [7, -19], [20/41, 21/43]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0095/)

Integer · qf-density13-integer-rational-grid-split · candidate





## 96. Rational Candidates Split by Integer Eligibility

Ordered representations=[8/1, 40/-2, 21/43, -22/-45] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.

View answer and explanation([Yes, Yes, No, No], [8, -20], [21/43, 22/45])

Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [8, -20], [21/43, 22/45]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0096/)

Integer · qf-density13-integer-rational-grid-split · candidate





## 97. Rational Candidates Split by Integer Eligibility

Ordered representations=[9/1, 42/-2, 22/45, -23/-47] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.

View answer and explanation([Yes, Yes, No, No], [9, -21], [22/45, 23/47])

Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [9, -21], [22/45, 23/47]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0097/)

Integer · qf-density13-integer-rational-grid-split · candidate





## 98. Rational Candidates Split by Integer Eligibility

Ordered representations=[10/1, 44/-2, 23/47, -24/-49] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.

View answer and explanation([Yes, Yes, No, No], [10, -22], [23/47, 24/49])

Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [10, -22], [23/47, 24/49]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0098/)

Integer · qf-density13-integer-rational-grid-split · candidate





## 99. Rational Candidates Split by Integer Eligibility

Ordered representations=[11/1, 46/-2, 24/49, -25/-51] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.

View answer and explanation([Yes, Yes, No, No], [11, -23], [24/49, 25/51])

Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [11, -23], [24/49, 25/51]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0099/)

Integer · qf-density13-integer-rational-grid-split · candidate





## 100. Rational Candidates Split by Integer Eligibility

Ordered representations=[12/1, 48/-2, 25/51, -26/-53] all have nonzero denominators. Give (integer-membership Yes/No list, integer values, reduced noninteger values), preserving original order within groups.

View answer and explanation([Yes, Yes, No, No], [12, -24], [25/51, 26/53])

Normalize to a positive denominator; denominator one means an integer, including zero and negative integers. Negative input denominators remain valid. The exact result, retaining the requested order, is ([Yes, Yes, No, No], [12, -24], [25/51, 26/53]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261009-advance-20261013-r02-integer-0100/)

Integer · qf-density13-integer-rational-grid-split · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261009-advance-20261013-r02-integer-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261009-advance-20261013-r02-integer-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261009-advance-20261013-r02-integer-foundations/validation-report.json)