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by surbhi arora - Wednesday, 10 September 2014, 05:38 AM
  Hi

Can someone please help me out with the following question ?

In how many ways can a pair of natural numbers be selected from a set containing the first 91 natural numbers such that their sum is a multiple of 13?

I followed the following approach -

a + b = 13 ,since a and b can not be zero ,they will take the value atleast 1. No of solutions = 11 + 2 - 1 C 2 -1 = 12
a + b = 26 , will have 25 - 1 solutions since we can not take 13,13
a + b = 39 ---- 38 solutions
a + b = 52 ----- 50 solutions
a + b = 65 ----- 62 solutions
a + b = 78 ----- 76 solutions
a + b = 91 ------ 90 solutions

Total = 12 + 24 + 38 + 50 + 62 + 76 + 90 = 352

But the answer is 315 , I cant figure out what am I doing wrong , please advise .