2015 · Q23
Let x and y be positive integers such that x > y. The expressions 3x+2y and 2x+3y when divided by 5 leave remainders 2 and 3 respectively. What is the remainder when (x-y) is divided by 5 ?
Elementary MathematicsSimple and Compound Interest
- A4
- B2
- C1
- D0
Explanation
This tests modular arithmetic. We are given 3x + 2y ≡ 2 (mod 5) and 2x + 3y ≡ 3 (mod 5). Subtracting the two congruences: (3x + 2y) - (2x + 3y) = x - y, and 2 - 3 ≡ -1 ≡ 4 (mod 5). So x - y leaves remainder 4 on division by 5. The condition x > y guarantees x - y is positive, so the remainder is genuinely 4 rather than a negative residue. Hence option (a).