Prove that if number a+4b is divisible by 13, then 10a+b is divisible by 13

Answer 1

Since a+4b is divisible by 13 we may write a+4b=13k for some integer k. But then we may write 10a+b=10(a+4b)-40b+b=13(10k)-39b=13(10k-3b) which means that the number 10a+b is a multiple of 13 and the proof is complete.