AIME 2000 I · 第 1 题

Solution

If a factor of $10^{n}$ has a $2$ and a $5$ in its prime factorization, then that factor will end in a $0$. Therefore, we have left to consider the case when the two factors have the $2$s and the $5$s separated, so we need to find the first power of 2 or 5 that contains a 0.

For $n = 1:$

$$2^1 = 2 , 5^1 = 5$$ $n = 2:$

$$2^2 = 4 , 5 ^ 2 =25$$ $n = 3:$

$$2^3 = 8 , 5 ^3 = 125$$ and so on, until,

$n = 8:$ $2^8 = 256$ | $5^8 = 390625$

We see that $5^8$ contains the first zero, so $n = \boxed{008}$.

Video Solution

https://www.youtube.com/watch?v=6cwg9DZ7bX4