Past fortnightly questions

This question involves substituting numbers into functions, simultaneous equations, and a bit of very basic number theory - for example, if $p$ and $q$ are both integers then so is $p+q$.

If you have not met it already, the formula for the triangular numbers is $\frac{n(n+1)}2$.  This is always an integer as we know that the triangular numbers are integers. Alternatively you could argue that $n$ and $n+1$ are consecutive numbers so one of them is even, and so has a factor of two.  Hence $\frac{n(n+1)}2$ must be an integer.