1. The problem statement, all variables and given/known data Theorem if n is a non-negative integer then 11 is a factor of 2^(4n+3)+3(5^n) Basic ideas if 2^(4n+3) =11k -3(5^n) then, 2^(4n+7) =11(16k -3(5^n))-3(5^(n+1)) 2. Relevant equations 3. The attempt at a solution So I understand how they get the result 2^(4n+3) =11k -3(5^n) but I have no idea how to obtain the second result, namely, 2^(4n+7) =11(16k -3(5^n))-3(5^(n+1)) ???? Im stumped I can obviously see how they have added another 2^4 but I really dont see how that produces RHS. Help!