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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Integer · qf-density13-integer-rational-grid-split · candidate
Canonical question JSONL · English JSONL · Validation report