When M is divided by 6, it leaves remainder 2, and when N is divided by 6, it…
2024
When M is divided by 6, it leaves remainder 2, and when N is divided by 6, it leaves remainder 3. What will be the remainder when M − N is divided by 6?
Answer: C. 5 — ConceptFor a positive divisor d, every integer has a unique least nonnegative remainder r with 0 ≤ r < d. Congruent integers differ by a multiple of d, so…
- A.
2
- B.
3
- C.
5
- D.
7
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Correct answer: C
Concept
For a positive divisor d, every integer has a unique least nonnegative remainder r with 0 ≤ r < d. Congruent integers differ by a multiple of d, so residues may be subtracted and then normalized into this range.
Application
Write M = 6a + 2 and N = 6b + 3 for some integers a and b.
Subtract: M − N = (6a + 2) − (6b + 3) = 6(a − b) − 1.
The term 6(a − b) is divisible by 6, so M − N has the same residue as −1 modulo 6.
Normalize −1 by adding one multiple of 6: −1 + 6 = 5. Therefore the least nonnegative remainder is 5.
Cross-check
Choose representatives M = 8 and N = 3. They have remainders 2 and 3 modulo 6, respectively; M − N = 5, whose remainder on division by 6 is 5.
Result
The remainder when M − N is divided by 6 is 5.