The question is that when positive integer n is divided by 5, the remainder is 1. When n is divided by 7, the remainder is 3. What is the smallest positive integer k such that k+n is a multiple of 35? The answer is 4.

I get that n= 7q +3 and it also equals n = 5p +1. I also know that you can make it to where 5p= 7q +2, but this is where I am really stuck as where do I go from there. I don't understand it when the book provides the explanation. I looked online and on videos and on those videos it said to find LCM, but why??