Proof by induction (two variables)

Question

$$ \text {Proof by induction, that for every natural number } $$$$n⩾90 \text { exists natural numbers }x_n \text{ and }y_n \text{ such that } n=7⋅x_n+16⋅y_n.$$

I really don't know how to do the example with two variales... Do you have any tips?

Answer 1

You aren't doing induction on the $x$ or the $y$; just on the index $n$.

For $n= 90$

You state $x_{90}= 6$ and $y_{90} = 3$ so $7*6 + 16*3 = 90$.

You then show that if for $n= k$ that $90 = 7x_k + 16y_k$ that

There is some $x_{k+1} = $ some modification of $x_k$ and $y_k$ and $y_{k+1}=$ some modification of $x_k$ and $y_k$.

Hint: Let $x_{k+1} = x_k +i$ and $y_{k+1}= y_k +j$. Then $90 = 7x_{k+1} + 16y_{k+1} = 7(x_k + i) + 16(y_k + j) = 7x_k + 16y_k + (7i + 16j) = 90 + (7i+16j)$. What can we say about $i$ and $j$?