The security guard problem

Solution 1:

Hint: How many divisors does $7$ have? How many divisors does $8$ have? How many divisors does $9$ have? What's special about $9$ and why?

Further Hint: If $a$ divides $n$, then $n = ab$ by definition. For example, $2$ divides $8$ because $8 = 2 \cdot 4$. In this way, divisors arise naturally in pairs.

If the numbers appearing in these pairs are all distinct, then there will be an even number of divisors. For example, the divisors of $8$ (in pairs) are $1$ and $8$ and also $2$ and $4$.

What about when the numbers in the pair aren't distinct? This can only happen if $n = a \cdot a$ for some $a$ (that is, if $n$ is a perfect square). For example, the divisors of $9$ (in pairs) are $1$ and $9$ and also $3$ and $3$. We don't count the $3$ twice, of course, so we get an odd number of divisors.