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A Biggy

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

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Why 24?

Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.

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Factoring a Million

In how many ways can the number 1 000 000 be expressed as the product of three positive integers?


Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

See the notes for information about why Lyndon likes this problem. He has also added a further related question (in green below).

If factorial $100$ ($100!$) was rewritten as the product of its prime factors how many $2$s and how many $5$s would there be?

Along the way confirm how many zeros are at the end of this large number and what the first digit to precede them is.