**Fall 2014 USA OMO Qn 9.** Let . How many zeros are at the end of the decimal representation of ?

To count the number of zeros at the end of , we need to find out the highest power of 2 that divides and the highest power of 5 that divides , then the number of zeros that the decimal representation of has is the lower of the two. Concretely, if for some integer such that , then the number of zeros in ‘s decimal representation is .

The short explanation above suggests that it would be a good idea to find the prime factorisation of . The key to the solution is realising that **each succeeding term is a multiple of the previous one, hence all succeeding terms are a multiple of the first one**. That is, we can write as

At this point, you might notice that for the second factor (the one in square brackets), all its terms except the 1 in front are divisible by 5. This means that the second factor will not be “contributing” any factors of 5 to the prime factorisation of . To simply the line above, we can write

for some integer . While factor might contribute some factors of 2 to the prime factorisation of , it won’t contribute any more factors of 5. **It should also be clear that the factor has more factors of 2 than factors of 5.** Hence,

.

There is a well-known formula that the highest power of a prime which divides is

Hence, number of ‘0’s at the end of is

The answer is **501**.

### Like this:

Like Loading...

*Related*