The product of an integer divisible by 3 and an even integer is divisible by 6.

We show that if x is an integer divisible by 3 and y is an even integer then xy is divisible by 6. Let x be divisible by 3 and let y be an even integer. Since x is divisible by 3, there is an integer a with x=3a. Since y is even, there is an integer b with y=2b. Observe that xy = (3a)(2b) = 6(ab). Hence there is an integer c, namely ab, with xy = 6c, showing y is divisible by 6.