Given: A five-digit number AABAA is divisible by 33.
When the number AABAA is divisible by 33 then its divisible by 3 and 11 because when a number say x is divisible by y, then x is divisible by the factors of y as well.
As AABAA is divisible by 3,we can say that sum of all the digits is a multiple of 3.
Therefore A + A + B + A + A = 0,3,6,9,12…
⇒ 4A + B = 0,3,6,9,12…
Now as AABAA is divisible by 11,then according to the divisibility rule of 11,the difference of sum of digits of odd places and even places should be divisible by 11.
Therefore, for the number AABAA, we can write the difference of sum of digits of odd places and even places as
(A + B + A)-(A + A) = 0,11,22,…
⇒ (2A + B)-(2A) = 0,11,22,..
⇒ B = 0(As B is a digit in the number AABAA,it can be only 0)
So, A can be any digit like 3,6,9,…and B = 0.
Hence the numbers in the form of AABAA divisible by 33 are
33033,66066 and 99099.
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