Factorials

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.

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

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!?

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

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

How many zeros does 50! have at the end?

Weekly Problem 17 - 2010
The value of the factorial $n!$ is written in a different way. Can you work what $n$ must be?

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

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

Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?