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

1 Answer
Will
Jul 17, 2016

27

Explanation:

The definition of an even number is #2k# and for odd #2k+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 #2m,2n# are even then #2m * 2n = 4mn =2(2mn)#.

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=2m, y=2n+1# and # (2m)(2n+1)=4nm+2m = 2(2nm+m)# 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 #(2m+1) * (2n+1) = 4mn+2n+2m+1 = 2(2mn+m+n)+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

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