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Challenge level: Challenge Level:1 Challenge Level:2 Challenge Level:3

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Broad Topics > Numbers and the Number System > Factorials

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Fac-finding

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Lyndon chose this as one of his favourite problems. It is accessible but needs some careful analysis of what is included and what is not. A systematic approach is really helpful.

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Weekly Challenge 36: Seriesly

Stage: 5 Short Challenge Level: Challenge Level:1

Prove that k.k! = (k+1)! - k! and sum the series 1.1! + 2.2! + 3.3! +...+n.n!

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How Does Your Function Grow?

Stage: 5 Challenge Level: Challenge Level:1

Compares the size of functions f(n) for large values of n.

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The Harmonic Triangle and Pascal's Triangle

Stage: 5

The harmonic triangle is built from fractions with unit numerators using a rule very similar to Pascal's triangle.

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Factorial Fun

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

How many divisors does factorial n (n!) have?

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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 since (6!) / (2^4 ) = 45. What is the highest power of two that divides exactly into 100!?

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Factoring Factorials

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

Find the highest power of 11 that will divide into 1000! exactly.

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Factorial

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

How many zeros are there at the end of the number which is the product of first hundred positive integers?

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Forgotten Number

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

I have forgotten the number of the combination of the lock on my briefcase. I did have a method for remembering it...

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Seriesly

Stage: 5 Challenge Level: Challenge Level:1

Prove that k.k! = (k+1)! - k! and sum the series 1.1! + 2.2! + 3.3! +...+n.n!

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N000ughty Thoughts

Stage: 4 Challenge Level: Challenge Level:1

How many noughts are at the end of these giant numbers?

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Growing

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)