13. This year, 2016, can be written as the sum of two positive integers $p$ and $q$ where $2 p=5 q$ (as $2016=1440+576)$. How many years between 2000 and 3000 inclusive have this property?
13. 143 For a year $Y$ to be expressible as as the sum of two positive integers $p$ and $q$ where $2 p=5 q$ we require $p+q=Y$ and $2 p=5 q$. From the first of these, it follows that $2 p+2 q=2 Y$ and hence $5 q+2 q=2 Y$.
Therefore $7 q=2 Y$ from which it follows that $Y$ is also divisible by 7 (since 2 and 7 are coprime).
We observe that $q=\frac{2 Y}{7}$ will be an integer less than $Y$ for all $Y$ that are multiples of 7 .
Then $p=Y-q$ will also be an integer.
We now must count all the multiples of 7 between 2000 and 3000 inclusive.
Since $1995=285 \times 7$ and $2996=428 \times 7$ there are $428-285=143$ multiples of 7 between 2000 and 3000 and hence there are 143 such years.