Mathematics Problem of the Week

We have twelve numbers, so $n=12$ & the mean is $$\stackrel{\_}{x}=\frac{\Sigma x_i}{12}=5$$ $$\Sigma x_i=60$$ The median is five too but as there are an even number of elements the mode is the mean of the sixth & seventh element when listed in order, so $$\mathrm{median}=\frac{x_6+x_7}{2}=5$$ $$x_6+x_7=10$$

Furthermore, the mode is also five, so both $x_6$ and $x_7$ are five, perhaps more. The range is eight, so

$$x_{\max }-x_{\min }=8$$

If the numbers are ordered this becomes

$$x_{12}-x_1=8$$

Knowing the $P(X=\mathrm{prime})=\mathrm{0.5}$ so there must be six prime numbers (some are repeated, eg. five)

$$x_1+x_2+x_3+x_4+x_5+5+5+x_8+x_9+x_{10}+x_{11}+x_1+8=60$$ $$2x_1+x_2+x_3+x_4+x_5+x_8+x_9+x_{10}+x_{11}=42$$