There are several different notions of average: the mean, the median, the mode and the range (see below for the definitions). If you work out each of these for the set of numbers 2, 5, 5, 6, 7, you'll notice something interesting — they are all equal to 5!

Can you find other sets of five positive whole numbers where mean = median = mode = range?

How many sets of five positive whole numbers are there with mean = median = mode = range = 100?

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### Definitions

Given a list of $n$ numbers $x_1,$ $x_2,$ $x_3,$ up to $x_n,$ (ordered by size) the*mean*is $$\frac{x_1+ x_2 + ... x_n}{n}.$$ The

*median*separates the lower half of the list from the higher half. It is the middle number if there are an odd number of numbers in the list, or the number half-way between the two middle numbers if there are an even number. So, assuming that $x_1 \leq x_2 \leq x_3 ... \leq x_n.$ The median is $$x_{(n+1)/2}$$ if $n$ is odd and $$\frac{x_{n/2} + x_{n/2+1}}{2}$$ if $n$ is even. The

*mode*is the number in the list that occurs most often — which means that there can be more than one mode. The

*range*is the difference between the largest and the smallest number in the list: $$x_n - x_1.$$