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If we divide the number $111222333444555666777888999$ by $111$.In which way one can find the number of digits of the result

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will this help:… – user59671 Feb 27 '13 at 16:46
Isn't it obviously 25? – MJD Feb 27 '13 at 16:47
9 digits representing 1-9 and then 16 zeros for a total of 25 digits. – JB King Feb 27 '13 at 17:11
up vote 1 down vote accepted

$n=111222333444555666777888999>111000000000000000000000000$ so $\frac{n}{111}>1\cdot10^{24}$ which has 25 digits.

On the other hand, $111>100$, so $\frac{n}{111}<\frac{n}{100}$ which has 25 (non-fractional) digits.

All in all, the number of digits $d$ satisfies $25\le d \le 25$ so $d=25$.

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Write $111222333444555666777888999$ as $\sum_{k=1}^9(10 - k) \times 111 \times 10^{3(k-1)}$ dividing by $111$ you get $\sum_{k=1}^9(10 - k) \times 10^{3(k-1)}$ and you get the largest value (the rest are smaller number, the sum does not change the number of digits) on this sum is $1 \times 10^{ 3 \times 8}$ that is $24$ zeros after $1$ which you get $25$.

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