# How many ways can you roll a pair of dice and get an even product?

Jul 17, 2016

27

#### Explanation:

The definition of an even number is $2 k$ and for odd $2 k + 1$. We know that a dice has 6 sides and when there are 2 there is a possible of ${6}^{2} = 36$ outcomes. This corresponds to each side from die 1 with each of the 6 sides from die 2.

In this case we are interested in only even outcomes.

If die 1 is even and die 2 is even then it will be even because if $2 m , 2 n$ are even then $2 m \cdot 2 n = 4 m n = 2 \left(2 m n\right)$.

If die 1 is even and die 2 is odd the the product is even because if x is even and y is odd then let $x = 2 m , y = 2 n + 1$ and $\left(2 m\right) \left(2 n + 1\right) = 4 n m + 2 m = 2 \left(2 n m + m\right)$ which is even. The same is true if die 1 is odd and die 2 is even.

So if die 1 is even then all the 6 sides from die 2 will be even. If die one is odd then all the even numbers of die 2 are even.

The last case is if both dice are odd. It turns out that this too is odd because $\left(2 m + 1\right) \cdot \left(2 n + 1\right) = 4 m n + 2 n + 2 m + 1 = 2 \left(2 m n + m + n\right) + 1$

So for 2,4,6 all 6 from two are even and for 1,3,5 only 2,4,6 are even thus 3×6 +3×3=27 even outcome