Question

Question 48

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, as if a number a is divisible by b, then it is also divisible by each factor of b.
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, ....
or 4A + B = 0, 3, 6, 9, ...................(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.