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 x times y is divisible by 6.


  1. Let x be divisible by 3 and let y be an even integer.

  2. Since x is divisible by 3 we know (Definition 1.2) that there is an integer a with x=3a.

  3. Since y is even, we know by Definition 1.1 that 2|y.

  4. Hence, by Definition 1.1, there is an integer b with y=2b.

  5. Observe that xy = (3a)(2b) = 6(ab).

  1. Hence there is an integer c, namely ab, with xy = 6c.

  2. Therefore, by Definition 1.1, y is divisible by 6.