Question

If n is a natural number, then which of the following is not always divisible by 2? (1) n2 – n (2) n3 – n (3) n2 – 1 (4) n3 – n2

Answer 1

Let us see these options one by one

1) n²-n=n(n-1)

This is a product of two consecutive numbers, so one of them must be even so this is always divisible by 2

2)n³-n=n(n²-1)=n(n+1)(n-1)

This is  the product of three consecutive numbers, so atleast one of them must be even,so this is also always divisible by 2

3)n²-1=(n+1)(n-1)

This one could be product of two even numbers ,or product of tw odd numbers too...so this one may or maynot be divisible by 2

4)n³-n²=n.n(n-1),which also involves the product of two consecutive numbers so is always divisible by 2

So the answer is (3)

Answer 2

ANSWER

Out of n and n+2, one is divisible by 2 and the other by 4, hence n(n+2) is divisible by 8.

Also n,n+1,n+2 are three consecutive numbers, hence one of them is divisible by 3.

Hence, n(n+1)(n+2) must be divisible by 24. This will be true for any even number n