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Broad Topics > Numbers and the Number System > Factorials

Fac-finding

Stage: 4 Challenge Level:

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.

Weekly Challenge 36: Seriesly

Stage: 5 Short Challenge Level:

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

How Does Your Function Grow?

Stage: 5 Challenge Level:

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

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.

Factorial Fun

Stage: 5 Challenge Level:

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

Powerful Factorial

Stage: 3 Challenge Level:

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!?

Factoring Factorials

Stage: 3 Challenge Level:

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

Factorial

Stage: 4 Challenge Level:

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

Forgotten Number

Stage: 3 Challenge Level:

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

Seriesly

Stage: 5 Challenge Level:

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

Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the. . . .

Growing

Stage: 5 Challenge Level:

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