24. The non-zero digits $p, q$ and $r$ are used to make up the three-digit number ' $p q r$ ', the two-digit number ' $q r$ ' and the one-digit number ' $r$ '. The sum of these numbers is 912 . What is the value of $q$ ?
A 3
B 4
C 5
D 6
E 0
Solution
C

The three-digit number ' $p q r$ ' is equal to $100 p+10 q+r$ and the two-digit number ' $q r$ ' is equal to $10 q+r$. Hence the sum of ' $p q r$ ', ' $q r$ ' and ' $r$ ' is equal to $100 p+20 q+3 r$. Since this sum is $912,3 r$ ends in ' 2 ' and hence, as $r$ is a single digit, $r=4$. Therefore $100 p+20 q=900$ and hence $10 p+2 q=90$. Therefore $2 q$ ends in ' 0 ' and, as the question tells us $q$ is non-zero. $q=5$.
It is then easy to work out that $p=8$ to show that a full solution to the question does exist.