17. What is the smallest three-digit positive integer which can be written in the form $p q^{2} r$, where $p, q$ and $r$ are distinct primes?
SolUtion
126

We consider multiples of $q^{2}$, for increasing values of $q$.
For $q=2$, we consider numbers of the form $4 k$, where $k$ is odd (since neither $p$ nor $r$ can then be 2 ).
$100=2^{2} \times 5^{2} ; 108=2^{2} \times 3^{3} ; 116=2^{2} \times 29 ; 124=2^{2} \times 31 ; 132=2^{2} \times 3 \times 11$, so 132 is the first multiple of $2^{2}$ of the required form.
For $q=3$ we consider multiples of 9 .
$108=2^{2} \times 3^{3} ; 117=3^{2} \times 13 ; 126=2 \times 3^{2} \times 7$, so 126 is the first multiple of $3^{2}$ of the required form.
For $q=5$ we consider multiples of 25 .
$100=2^{2} \times 5^{2} ; 125=5^{3}$; and subsequent multiples are larger than 126 .
For $q=7$ we consider multiples of 49 . However, the smallest three-digit multiple, 147, is already larger than 126 .
For $q=11$ we consider multiples of 121 . However, $121=11^{2}$ and subsequent multiples are larger than 126 .
For all other possible values of $q, q^{2}$ is larger than 126. Therefore, 126 is the smallest such number.