### Old Nuts

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?

### Sixational

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. Prove that all terms of the sequence are divisible by 6.

### Telescoping Functions

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

# Factorial Fun

##### Age 16 to 18 Challenge Level:

We denote the product of the first 20 natural numbers by 20! and call this 20 factorial.

(a) What is the highest power of 5 which is a divisor of 20 factorial? Just how many factors does 20! have altogether?

(b) Show that the highest power of $p$ that divides $500!$, where $p$ is a prime number and $p^t < 500 < p^{t+1}$, is $$\lfloor 500/p\rfloor+\lfloor 500/p^2\rfloor+\dotsb+\lfloor 500/p^t\rfloor,$$ where $\lfloor x\rfloor$ (the floor of $x$) means to round down to the nearest integer.  (For example, $\lfloor 3\rfloor=3$, $\lfloor 4.7\rfloor=4$, $\lfloor -2.7\rfloor=-3$, and so on.)

(c) How many factors does $n!$ have?