Question

The pair of two-digit numbers 12 and 63 has the interesting property that if the product expression $12\times 63$ is read in reverse order $36\times 21$, the two product expressions are equal, i.e. $12\times 63=36\times 21$. How many pairs of two-digit numbers with this property are there in which one of the numbers in the pair is greater than 20 and less than 30, and the units' digit of the other number in the pair is an odd prime?___

Answer 1

(20 + x)(10y + p) = (10p + y)(10x + 2) $\Rightarrow$ px = 2y. For p = 5 or 7 $\Rightarrow$ y = p and x = 2, and for p = 3 $\Rightarrow$ y = 3, 6 or 9 and x = 2, 4 or 6 respectively.