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Counting Factors

Is there an efficient way to work out how many factors a large number has?

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Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

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Oh! Hidden Inside?

Find the number which has 8 divisors, such that the product of the divisors is 331776.

Coin Collection

Stage: 3 Short Challenge Level: Challenge Level:2 Challenge Level:2
The number of coins in Tom's collection is 3 more than a multiple of 6 and also 7 more than a multiple of 8. The smallest numbers that satisfies both conditions is 15. The lowest common multiple of 6 and 8 is 24, so the conditions will also be met by numbers that exceed 15 by a multiple of 24, that is 39, 63, 87, etc. So when Tom puts his coins in piles of 24, 15 remain.

This problem is taken from the UKMT Mathematical Challenges.
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