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N000ughty Thoughts

Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the number of noughts in 10 000! and 100 000! or even 1 000 000!

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Forgotten Number

I have forgotten the number of the combination of the lock on my briefcase. I did have a method for remembering it...

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How many zeros are there at the end of the number which is the product of first hundred positive integers?

Trailing Zeros

Stage: 3 and 4 Short Challenge Level: Challenge Level:2 Challenge Level:2


Every zero at the end of a number corresponds to one of the factors being 10 (e.g., 23000 has 3 factors of 10 (i.e. $10^3$ divides 23000). But 10 itself can be factorised into $5 \times 2$, so we're looking for the smallest power of 2 or 5 and that will be our answer. $50!$ has at least 25 factors of 2, plenty more than we expect it to have factors of 5. Of the numbers 1 to 50, ten of them (5, 10, ..., 50) have 5 as a factor, and two of those have two factors of 5. So $50!$ has 12 factors of 5 (less than 25, good). Hence $50!$ has 12 zeros at its end.

This problem is taken from the UKMT Mathematical Challenges.
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