A2. A palindrome is a whole number whose digits are the same when read from left to right as from right to left. For example, 565 and 7887 are palindromes. Find the smallest six-digit palindrome divisible by 12 .
Solution: We need a number in the form $\overline{a b c c b a}$ that is divisible by both 3 and 4 . Thus, we require that the sum of digits $2 a+2 b+2 c$ be divisible by 3 , and that the number formed by the last two digits, $10 b+a$ be divisible by 4 . Due to the second condition, it is impossible to have $a=1$, so the smallest number must have $a \geq 2$. If $a=2$, then we cannot have $b=0$, again by the second condition, so we must have $b \geq 1$. The number 210012 satisfies both conditions, so it is the smallest six-digit palindrome divisible by 12 .
Answer: 210012 .