7. In 2021 , a first class postage stamp cost 85 p and a second class postage stamp cost 66 p. In order to spend an exact number of pounds and to buy at least one of each type, what is the smallest total number of stamps that should be purchased?
A 10
B 8
С 7
D 5
E 2
Solution
C
The cost of $r$ first class postage stamps at 85 p each, and $s$ second class postage stamps at 66p each is $(85 r+66 s)$ p.

This is an exact number of pounds provided that
$$
85 r+66 s=100 t
$$
for some positive integer $t$.
We therefore seek the solution of (1) in which $r, s$ and $t$ are positive integers with $r+s$ as small as possible.

Equation (1) may be rearranged as
$$
85 r=100 t-66 s .
$$

Since 100 and 66 are both divisible by 2 , it follows from (2) that $85 r$ is divisible by 2 . Therefore $r$ is divisible by 2 . Hence $r \geq 2$.

Equation (1) may also be rearranged as
$$
66 s=100 t-85 r .
$$

Since 100 and 85 are both divisible by 5 , it follows from (3) that $66 s$ is divisible by 5 . Therefore $s$ is divisible by 5 . Hence $s \geq 5$.

We now note that when $r=2$ and $s=5$, we have
$$
85 r+66 s=85 \times 2+66 \times 5=170+330=500 .
$$

Therefore equation (1) has the solution $r=2, s=5$ and $t=5$. Because $r=2$ and $s=5$ are the least possible values for $r$ and $s$, it follows that $r+s$ has the least possible value among the solutions of (1) in which $r, s$ and $t$ are positive integers.

Since, in this case, $r+s=2+5=7$, we deduce that the least number of stamps that should be purchased is 7 .

For INVESTIGATION
7.1 Today a first class postage stamp costs 95 p, and a second class postage stamp costs 68 p. In order to spend an exact number of pounds and to buy at least one of each type, what is the smallest total number of stamps that should be purchased?
7.2 Find the solution of the equation
$$
45 r+56 s=100 t
$$
in which $r, s$ and $t$ are positive integers and $r+s$ is as small as possible.