x and y are two numbers which, when divided by 6, leave remainders of 4 and 5…

Concept: When x leaves remainder p and y leaves remainder q on division by n, the remainder of (x + y) on division by n equals the remainder of (p + q) on division by n — because x = na + p and y = nb + q, so x + y = n(a + b) + (p + q), and only the (p + q) portion needs to be reduced modulo n.

1. Write x = 6a + 4 and y = 6b + 5 for integers a, b, since x leaves remainder 4 and y leaves remainder 5 on division by 6.

2. Add the two expressions: x + y = 6a + 4 + 6b + 5 = 6(a + b) + 9.

3. Reduce the constant term modulo 6: 9 = 6(1) + 3, so x + y = 6(a + b + 1) + 3.

4. The remainder when x + y is divided by 6 is therefore 3.

Cross-check: take a = b = 0, so x = 4 and y = 5. Then x + y = 9, and 9 divided by 6 leaves remainder 3 — confirming the result above.