A6. A two-digit number ' $a b$ ' is multiplied by its reverse ' $b a$ '. The ones (units) and tens digits of the four-digit answer are both 0 .
What is the value of the smallest such two-digit number ' $a b$ '?
Solution
Since the units digit of the answer is a zero, either $a$ or $b$ must be 5 . Without loss of generality, assume $a=5$. Therefore, $b$ is even.
Since the answer ends in ' 00 ', it is a multiple of 100 and hence is a multiple of 25 . Therefore, since $b \neq 0$ and ' $b a$ ' ends in 5 , ' $b a$ ' has a factor of 25 . The only 2-digit multiples of 25 ending in 5 are 25 and 75 . However 7 is not even. Therefore ' $b a$ ' $=25$ and the smallest two-digit number is 25 .