22. Last week Evariste and Sophie both bought some stamps for their collections. Each stamp Evariste bought cost him $£ 1.10$, whilst Sophie paid 70p for each of her stamps. Between them they spend exactly $£ 10$. How many stamps did they buy in total?
A 9
B 10
C 11
D 12
E 13
Solution: D
Suppose that Evariste buys $x$ stamps and Sophie buys $y$ stamps. Then $1.10 x+0.70 y=10$ and hence $11 x+7 y=100$. We have one equation with two unknowns, so we need to use the fact that in this problem $x$ and $y$ are non-negative integers. As $y=\frac{100-11 x}{7}$, we seek a value of $x$ for which $100-11 x$ is divisible by 7 . Taking $x=0,1,2, \ldots, 9$ we have $100-11 x=100,89,78,67,56,45,34,23,12,1$. We see that only for $x=4$ is $100-11 x$ divisible by 7. So $x=4$ and $y=\frac{56}{7}=8$. So Sophie buys 8 stamps and Evariste buys 4 stamps, and they buy 12 stamps in total.