, 2014

Suppose that the number of 2  stamps that Sonita buys is $x$.

Then the number of 1  stamps that she buys is $10x$.

The total value of the 2  and 1  stamps that she buys is $2(x)+1(10x)=12x$ .

Since she buys some 5  stamps as well and the total value of the stamps that she buys is 100 , then the value of the 5  stamps that she buys is $(100-12x)$ .

Thus, $100-12x$ must be a multiple of 5. Since $100$ is a multiple of 5, then $12x$ must be a multiple of 5, and so $x$ is a multiple of 5 (since 12 has no divisors larger than 1 in common with 5).

Note that $x>0$ (since she buys some 2  stamps) and $x<9$ (since $12x$ is less than 100). The only multiple of 5 between 0 and 9 is 5, so $x=5$.

When $x=5$, the value of 5  stamps is $100-12x = 100-12(5)=40$  and so she buys $\frac{40}{5}=8$ 5  stamps.

Finally, she buys 5 2  stamps, 50 1  stamps, and 8 5  stamps, for a total of $5+50+8=63$ stamps.

(Checking, these stamps are worth $5(2)+50(1)+8(5)=10+50+40=100$  in total, as required.)