Which of the following numbers is divisible by 24?

Concept:

A number is divisible by a composite number that factors into two coprime numbers p and q (gcd(p, q) = 1) exactly when it is divisible by both p and q individually — since p and q share no common factor, passing both tests guarantees divisibility by their product. Here, 24 = 3 × 8, and gcd(3, 8) = 1, so a number is divisible by 24 only if its digit sum is divisible by 3 AND its last three digits form a number divisible by 8.

Application:

1. 35718: digit sum = 3+5+7+1+8 = 24, divisible by 3; last three digits 718, and 718 ÷ 8 = 89.75 — not divisible by 8, so it fails.

2. 63810: digit sum = 6+3+8+1+0 = 18, divisible by 3; last three digits 810, and 810 ÷ 8 = 101.25 — not divisible by 8, so it fails.

3. 537804: digit sum = 5+3+7+8+0+4 = 27, divisible by 3; last three digits 804, and 804 ÷ 8 = 100.5 — not divisible by 8, so it fails.

4. 3125736: digit sum = 3+1+2+5+7+3+6 = 27, divisible by 3; last three digits 736, and 736 ÷ 8 = 92 exactly — divisible by 8. Both conditions hold.

Cross-check:

Dividing directly confirms it: 3125736 ÷ 24 = 130239, an exact integer with no remainder.

So 3125736 is the number divisible by 24.