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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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A number N is divisible by 10, 90, 98 and 882 but it is NOT divisible by 50 or 270 or 686 or 1764. It is also known that N is a factor of 9261000. What is N?

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The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?

Weekly Problem 14 - 2008

Stage: 3 and 4 Challenge Level: Challenge Level:1

As the product of each pair has the same value, this value must be product of the smallest and largest numbers, that is $5\times72$.

So the number which is paired with $10$ is $5\times72\div10$, that is $36$.

This problem is taken from the UKMT Mathematical Challenges.

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