x(x + 2)(x + 3)

We can analize 1 by 1 the answer choices

A) x is odd implies that y is odd ?

A fast way is to plug value... So, x=1

1*(1 + 2)*(1+3) = 12 = even

Wrong.

B) even only when x is even ?

We saw just before that if x is 1, we have an even. So no.

Wrong.

C) even only when x is odd ?

If x=0, then 0*(0+2)*(0+3) = 0 = even

Wrong.

D) divisible by 3 when x is odd ?

If x=5, 5*(5 + 2)*(5 + 3) = 5*7*2*2*2.... Not divisible by 3

Wrong.

E) divisible by 2 ?

Yes... By POE

... but, why not try to solve it?

Case 1 : x is odd and so x = 2*k+1 where k is an integer.

(2*k+1) * (2*k+1 + 2) * (2*k+1 + 3)

= (2*k+1) * (2*k+3) * (2*k+4)

= (2*k+1) * (2*k+3) * 2*(k+2)

= 2 * [(2*k+1) * (2*k+3) * (k+2)] ..... Divisible by 2 >>> OK

Case 2 : x is even and so x = 2*k where k is an integer.

(2*k) * (2*k + 2) * (2*k + 3)

= 2 * [(k) * 2 * (k+1) * (2*k+3)] ..... Divisible by 2 >>> OK