The product of four consecutive natural numbers is always divisible by:

Concept: For any n consecutive natural numbers, their product is always divisible by n! (n factorial), because among n consecutive integers there is always a complete set of factors guaranteeing this. For n = 4, 4! = 4 × 3 × 2 × 1 = 24.

Application:

1. Take the smallest case: 1, 2, 3, 4. Product = 1 × 2 × 3 × 4 = 24.

2. This is divisible by 24 (24 ÷ 24 = 1), but NOT by 30, 45, or 56 (none of these divide 24 evenly).

3. So 30, 45, and 56 already fail on this minimal case and cannot be universal divisors of the product of four consecutive natural numbers.

Cross-check: Try a second set: 2, 3, 4, 5. Product = 2 × 3 × 4 × 5 = 120. 120 ÷ 24 = 5, a whole number, confirming 24 continues to divide the product. (Note 120 also happens to be divisible by 30, but the first case already ruled 30 out as a universal divisor — it must hold for every set of four consecutive numbers, not just one.)

Result: The product of four consecutive natural numbers is always divisible by 24.