### Forgotten Number

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

### Factoring Factorials

Find the highest power of 11 that will divide into 1000! exactly.

### Powerful Factorial

6! = 6 x 5 x 4 x 3 x 2 x 1. The highest power of 2 that divides exactly into 6! is 4 since (6!) / (2^4 ) = 45. What is the highest power of two that divides exactly into 100!?

# Weekly Problem 13 - 2009

##### Stage: 3 Short Challenge Level:

12.

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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