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Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?


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


Lyndon chose this as one of his favourite problems. It is accessible but needs some careful analysis of what is included and what is not. A systematic approach is really helpful.

N000ughty Thoughts

Age 14 to 16
Challenge Level

Is it now well-known that 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!