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Question
A five-digit number AABAA is divisible by 33. Write all the numbers of this form.
Solution
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. 4 + 4 + 8 + A + 4 = 0, 3, 6, 9 ...
or 4/4 + 8 = 0, 3, 6, 9, ...(i)
From equation (i), we have
Further, the given number is also divisible by 11,
Therefore (2/4 + 8) – 2A = 0, 11, 22, ...
B = Q 11, 22, ...
8 = 0 ...[8 is a digit of the given number]
4/4 = 12 or 24 or 36 A = 3, 6, 9
Hence, the required numbers are 33033, 66066 and 99099.
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