Question

A five- digit number " AABAA " is divisible by 33. Write all the numbers of this form.

Answer 1

Given, a number of the form AABAA is divisible by 33. Then, it is also divisible by 3 and 11 which are factors of 33
Since, AABAA is divisible by 3, sum its digits is also divisible by 3.
i.e. A + A + B + A + A = 0, 3, 6, 9, 12, 15, 18,21,24, and so on...
or 4A + B = 0, 3, 6, 9, 12, 15, 18,21,24, and so on.... (i)

Further, the given number is also divisible by 11, therefore (2A + B) - 2A = 0, 11, 22, ...
$\Rightarrow$ B = 0, 11, 22, ...
$\Rightarrow$ B = 0 [$\because$ B is digit of the given number]
From Eq. (i), we have
4A = 12 or 24 or 36
$\Rightarrow$ A = 3, 6, 9
Hence, the required numbers are 33033, 66066, and 99099.