Question

in how many ways can three numbers be chosen from the set {1,2,3,...,12} such that their sum is divisible by 3.

Answer 1

Answer: Let’s call the set of integers  A  

A={1,2,3,4,5,6,7,8,9,10,11,12}  

Let’s define a new set  B , where  B[n]=A[n]mod3  

B={1,2,0,1,2,0,1,2,0,1,2,0}  

For the sake of clarity, I’m going to label each number with a subscript letter.

B={1a,2a,0a,1b,2b,0b,1c,2c,0c,1d,2d,0d}  

The question can now be rephrased as: How many different ways can you pick 3 distinct elements from  B  such that their elements sum to multiples of 3?

You should be able to see that there are now only these two cases.