**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 $100$?

The

Every logical solution to a problem has its own beauty.

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- Phlembac Adib Hasan
**Posts:**1016**Joined:**Tue Nov 22, 2011 7:49 pm**Location:**127.0.0.1-
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\[100=2^2.5^2\]

We first get $25$ at $25^{th}$ term.That term is also divisible by $2^2$.So the answer is $25^{th}$ term.

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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Can you explain this logically?

- nafistiham
**Posts:**829**Joined:**Mon Oct 17, 2011 3:56 pm**Location:**24.758613,90.400161-
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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

\[\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0\]

Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.

Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.