Let a and b be integers. Show that 29 divides 3a+2b if and only if it divides 11a+17b.

I am unable to solve this

  1. Am I correct in interpreting this as : "If 3a+2b is divisible by 29 then prove that 11a+17b is also divisible by 29." ?

  2. What is an intuitive way to solve this ?

P.S : Peter Winkler has released his book, Mathematical puzzles for free. Here is the link to it : https://math.dartmouth.edu/news-resources/electronic/puzzlebook/index.php


1 Answer 1


Let the number be $x$. Then, 29 divides $x$ iff 29 divides $18a+12b=6x$ iff 29 divides $11a+17b=29(a+b)-6x$.


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