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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.

Powerful Factorial

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Each number will appear as a factor in 100!. So, for example 100! = 100x99x98x...where 22is a factor of 100, 2 is a factor of 98 and 25 is a factor of 96. Are there patterns to the powers of the number that appear as you consider each term of the factorial? Can this pattern be generalised for other numbers?