# Maths item of the month

### April 2014

Divisibility of consecutive integers

Let *N* = (*n*+1)(*n*+2)...(2*n*), i.e. the product of *n* consecutive integers from *n*+1 to 2*n*.

Prove that, for any positive integer *n*, *N* is divisible by 2^{n} but not a higher power of 2.

For example, for *n* = 3:

*N* = 4×5×6

= 120.

120 is divisible by 8 but not divisible by 16.

### March 2014

Inner Circle

In the diagram various regular polygons, P, have been drawn whose sides are tangents to a circle, C.

Show that for any regular polygon drawn in this way:

### February 2014

Cones from a Circle

An angle *θ* is cut out of a circle of card to create two sectors: a major sector and a minor sector. The two sectors are then folded to make cones.

What angle *θ* is required to obtain the largest value for the sum of the volumes of the two cones?

### January 2014

An Unexpected Answer

Mr Student sets his class the following problem:

A committee of 3 students is to be chosen from a group of 13 students of which 8 are girls and 5 are boys. The students are selected at random, without replacement. What is the expected number of girls on the committee?

Anne Student immediately responds that the answer is .

She gives the reason that there are 3 students to be chosen and the proportion of girls is

so she calculated

Is she correct?

If the number of students was different would her method work?