Question

The product of three consecutive natural numbers is always a multiple of 3.

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Answer 1

Step-by-step explanation:

This part is an answer to the initial question (asking for the SUM instead of the Product)

Three consecutive natural numbers are n−1,n,n+1 and their sum is 3n. But if n is odd, this sum is not divisible by 6.

Answer related to the PRODUCT:

Product of three consecutive natural numbers is (n−1)n(n+1)=n3−n. Denote it P.

If n is even (n=2k), then n3=8k3 is even and P is even as the difference of two even numbers. Similarly, if n is odd, then n3 is odd and the difference of two odd numbers is even.

We conclude that P is even (divisible by 2).

For divisibility by 3, consider the 3 cases:

n≡0(mod3), then n3≡0(mod3), from where P≡0(mod3);

n≡1(mod3), then n3≡1(mod3), which gives P≡0(mod3);

n≡2(mod3), then n3≡2(mod3) and we have again P≡0(mod3).

Conclusion: P is divisible by 2 and by 3, therefore is divisible by 6.