Question

Prove that if $n$ is a positive even integer,then 24 divides $n(n+1)(n+2)$ .

Answer 1

for three consecutive integers one has to surely be multiple of 3.

Now if n is even it can be either of 4m or 4m+2 where m is also a number.

if it is 4m then n+2=4m+2 so 4m(4m+2)=8k(k is another number)

if it is 4m+2 then n+2=4m+2 so (4m+4)(4m+2)=8j(j is another number)

In any case we get 8 as factor and we deduced earlier that 3 is surely a factor

8 and 3 are co-prime.So it is a multiple of 8$\times$3=24.