A five-digit number AABAA is divisible by 33. Write all the numbe… - Vidyadip

Question

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

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Answer

Given, a number of the form $AABAA$ is divisible by $33.$
Then, it is also divisible by $3$ and $11$, as if a number ?
is divisible by ?, 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 $\frac{4}{4}+8 = 0, 3, 6 9, (i)$ From Eq. $(i),$
we have Further, the given number is also divisible by $11,$
therefore $\Big(\frac{\text{z}}{4}+8\Big)-2\text{A} = 0, 11, 22 B = Q 11, 22, 8 = 0 [V8$ is a digit of the given number] $\frac{4}{4}\ = 12$ or $24$ or $36 A = 3, 6 9$
Hence, the required numbers are $33033, 66066$ and $99099.$

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