Consider numbers of the form un = 1! + 2! + 3! +...+n!. How many such numbers are perfect squares?
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.
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 zeros are there at the end of the number which is the product of first hundred positive integers?
Find the highest power of 11 that will divide into 1000! exactly.
How many noughts are at the end of these giant numbers?
I have forgotten the number of the combination of the lock on my briefcase. I did have a method for remembering it...
An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.