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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

6! = 6 x 5 x 4 x 3 x 2 x 1

The highest power of 2 that divides exactly into 6! is 4. 

((6!) / (2 4 ) = 45)

What is the highest power of two that divides exactly into 100! (100 x 99 x 98 x 97 x ... x 1)?

What is the highest power of three that divides exactly into 100! ?


Can you see any patterns in the calculation of the highest powers of each number that divides exactly into 100!?

Can you generalise your findings to any factorial and any number?