Prove by induction that $7a + 10b$ can represent all integers $n \ge 54$

Question

How do you use proof by induction to prove that $7a + 10b$ can represent all integers $n \ge 54$ given that $a$ and $b$ are both positive integers?

Base Case is true because $7 \cdot 2 + 10 \cdot 4 = 14 + 40 = 54$. Then I don't know how to set up the induction step.

Answer 1

Considering the requirement that you must use induction. Assume a number $n >53$, and $n=7k1+10k2$.

Consider $N=7K1+10K2$ with $K1=k1+3$ and $K2=k2-2$ so that

$N=7(k1+3)+10(k2-2)=7k1+10k2+21-20=n+1$

You then need to show that there is a combination for 54 and that that k2>2 for this combination (=your base case) so that the factors are all positive.