## BdMO National 2012: Junior 10

Zzzz
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### BdMO National 2012: Junior 10

Problem 10:
The $n$-th term of a sequence is the least common multiple (l.c.m.) of the integers from $1$ to $n$. Which term of the sequence is the first one that is divisible by $100$?
Every logical solution to a problem has its own beauty.

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### Re: BdMO National 2012: Junior 10

$100=2^2.5^2$
We first get $25$ at $25^{th}$ term.That term is also divisible by $2^2$.So the answer is $25^{th}$ term.
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Arafat
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### Re: BdMO National 2012: Junior 10

Can you explain this logically?

nafistiham
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### Re: BdMO National 2012: Jounior 10

He means to divide $100$ the term must divide $2^2,5^5$ the first term that divides $25$ is the $25^{th}$ term and it also divides $4$.That's why
$\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0$
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