The $n-th$ term of a sequence is the least common multiple (LCM) of the integers from $1$ to $n$. Which term of the sequence is the first one that is divisible by $100$?
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Note: This is a problem from BDMO - $2012$
2026-03-26 17:33:38.1774546418
The $n-th$ term of a sequence is the LCM of the integers from $1$ to $n$
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For some prime $p$, the greatest $p^n$ that divides the LCM of the integers $1$ through $n$ is the greatest $p^n$ that is less than $n$. For example, for the LCM of the integers $1$ through $50$, the greatest $7^n$ that divides the LCM is $49$ since $7^2=49<50$, but $7^3=343>50$. $7^3$ can not divide the LCM of the integers $1$ through $50$ because there is no integer less than or equal to $50$ that has $7^3$ as a factor.
Thus, for the number $100=2^2*5^2$, we need the LCM to divide $2^2=4$ and $5^2=25$. The maximum of these two prime powers is $25$. Thus, the lowest integer $n$ such that the LCM of $1$ through $n$ is divisible by $100$ is $25$ since it is the smallest LCM that has $5^2$ as a factor.