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N000ughty Thoughts

How many noughts are at the end of these giant numbers?

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

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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.


Age 14 to 16 Challenge Level:

Consider numbers of the form $u_n = 1! + 2! + 3! +...+n!$.

How many such numbers are perfect squares?