Problem #PRU-100489

Problems Number Theory Divisibility Products and factorials The fundamental theorm of arithmetic. Prime factorisation.

Problem

Let us introduce the notation – we denote the product of all natural numbers from 1 to $n$ by $n!$ . For example, $5! = 1 \times 2 \times 3 \times 4 \times 5 = 120$ .

a) Prove that the product of any three consecutive natural numbers is divisible by $3! = 6$ .

b) What about the product of any four consecutive natural numbers? Is it always divisible by 4!=24?

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