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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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Repeaters

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.

What an Odd Fact(or)

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3
Can you show that $1^{99} + 2^{99} + 3^{99} + 4^{99} + 5^{99}$ is divisible by $5$?