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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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Rachel's Problem

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Age 14 to 16 Challenge Level:

Find the maximum of

$${1\over p} + {1\over q} + {1\over r}$$

where $p$, $q$ and $r$ are positive integers and

$${1\over p} + {1\over q} + {1\over r} < 1.$$

Prove that it is indeed a maximum.