Use induction: postage $\ge 64$ cents can be obtained using $5$ and $17$ cent stamps. I have come up with:
Assume for any $n\ge 64$ there exists numbers $x$ and $y$ such that $n = 17x + 5y$
and also then that $n+1 = 17x + 5y + 1$
but am fairly new to the concept of induction and not sure where to go after this.
 A: You can exhibit combinations of stamps for $64,65,66,67$ and $68$ cents as your base case, and then use the inductive step of adding a $5$ cent stamp for any greater amount.
The inductive phase addresses values of $69$ cents or more and can assume that the previous $5$ lesser values all have valid combinations of stamps.

More explicitly:
Base case:


*

*$n=64$: $2\times$ $17$c stamps and $6\times$ $5$c stamps:

*$n=65$: you can fill these in

*$n=66$: 

*$n=67$: 

*$n=68$: 


Inductive step: For any $k\ge 69$, assume that $n$ cents in the previous five values can be made by $a_n\times$ $17$c stamps and $b_n\times$ $5$c stamps.
Then for $k$ cent value set $a_k = a_{k-5}$ and $b_{k} = b_{k-5}+1$, giving:
$\begin{align}
17 a_k + 5 b_k &= 17a_{k-5} + 5(b_{k-5}+1)\\
 &= 17a_{k-5} + 5b_{k-5} + 5\\
 &= (k-5) + 5 = k\\
\end{align}$
as required
A: *

*Base step: Show that there are values of $x$ and $y$ that satisfy $17x+5y=64$.

*Inductive step: Assume that there exist $x$ and $y$ such that $17x+5y=n$. Use that assumption to show that there must exist $x'$ and $y'$ such that $17x'+5y'=n+1$.


Checking the base step is important and I will leave that to you. For the inductive step, using $17x+5y=n$ we can re-write $n+1=17x+5y+1$. Then, use the hint given by lab bhattacharjee.
A: Base case: show that it works for n = 64
17x + 5y = 64
$x = 2, y = 6$ [OK]

Inductive Step: supposing it works for $n$, prove it works for $n+1$
there must exist $x'$ and $y'$ such that:
$17x' + 5y' = n + 1$
using the hypothesis, we can say $n = 17x + 5y$
$17x' + 5y' = 17x + 5y + 1$
$17(x' - x) + 5(y' - y) = 1$
say $(x' - x) = a$ and $(y' - y) = b$
if there are $a$ and $b$ such that $17a + 5b = 1$, than there will exist satisfying $x'$ and $y'$.
$a = -2, \;b = 7 \implies 17(-2) + 5(7) = -34+35=1$
Then, as we know $x$ and $y$ exists by hypothesis, for $n+1$ you just need to use $x' = x - 2$ and $y' = y + 7$ and you will get it.
