20. For how many three-digit numbers can you subtract 297 and obtain a second three-digit number which is the original three-digit number reversed?
A 5
B 10
C 20
D 40
E 60
Solution
Suppose that ' $p q r$ ' is a three-digit number whose digits are reversed when 297 is subtracted. Since ' $p q r$ ' represents the number $100 p+10 q+r$ and ' $r q p$ ' represents $100 r+10 q+p$, we have $100 p+10 q+r-297=100 r+10 q+p$. This equation can be rearranged to give $99 p-99 r=297$ and hence we have $p-r=3$. Since we know that ' $r q p$ ' is a three-digit number, $r \neq 0$. Therefore there are six possibilities for the pair $(p, r)$, namely $(4,1),(5,2)$, $(6,3),(7,4),(8,5)$ and $(9,6)$. The middle digit, $q$, can be any one of the 10 digits. Therefore the number of possible values for the original three-digit number is $6 \times 10=60$.