A number is divided by 416 leaves remainder 112. What will be the remainder…

Concept: If a number N leaves remainder r when divided by d₁, and d₂ divides d₁ exactly, then N leaves the same remainder as r (reduced mod d₂) when divided by d₂ — because the multiple of d₁ inside N is automatically a multiple of d₂ and contributes nothing to that remainder.

Application:

1. Since the number N divided by 416 leaves remainder 112, write N = 416q + 112 for some whole number q.

2. Check whether 32 divides 416 exactly: 416 ÷ 32 = 13 with no remainder, so 416 = 32 × 13.

3. Because 416 is an exact multiple of 32, the term 416q (= 32 × 13q) is also an exact multiple of 32, so it leaves remainder 0 when N is divided by 32.

4. So the remainder of N on division by 32 is entirely determined by the remainder of 112 on division by 32.

5. Divide 112 by 32: 32 × 3 = 96, and 112 − 96 = 16, so 112 leaves remainder 16 when divided by 32.

6. Hence the remainder when N is divided by 32 is 16.

Cross-check: Pick a concrete value of q, say q = 1, so N = 416 × 1 + 112 = 528. Dividing 528 by 32: 32 × 16 = 512, and 528 − 512 = 16 — the same remainder, confirming the result holds for any q.