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Counting Factors

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

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.

Oh! Hidden Inside?

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

What an Odd Fact(or)

Age 11 to 14 Challenge Level:
 
Can you show that $1^{99} + 2^{99} + 3^{99} + 4^{99} + 5^{99}$ is divisible by $5$?