divisibility

If n is a positive integer and (n + 1)(n + 3) is odd, then show that (n + 2)(n + 4) must be a multiple of 8.




Answer:

since (n + 1)(n + 3) is odd, therefore, (n + 1) is odd and (n + 3) is odd.
this implies n is even 
so, we write n = 2k 

(n + 2)(n + 4) = (2k + 2)(2k + 4) = 2.(k + 1).2.(k + 2) = 4.(k + 1)(k + 2)

since (k + 1) & (k + 2) are consecutive numbers, one of them is always even.
so, 

(n + 2)(n + 4) is a multiple of 4 x even number = 8