Mean, Median and Mode of Integers

Sum of 6 integers = $15 \times 6 = 90$. Arranged in order $x_1 \le x_2 \le x_3 \le x_4 \le x_5 \le x_6$. Median = $\frac{x_3 + x_4}{2} = 18 \implies x_3 + x_4 = 36$. Mode < 18, so the mode must be $x_1 = x_2$. To maximize $x_6$, minimize $x_1, x_2, x_3, x_4, x_5$. Set $x_1 = x_2 = 1$. Since $x_3 + x_4 = 36$ and $x_3 \neq x_4$ (otherwise mode would be 18), choose $x_3 = 17, x_4 = 19$. To minimize $x_5$, set $x_5 = 20$ (since $x_5 > x_4 = 19$). Then $x_6 = 90 - (1 + 1 + 17 + 19 + 20) = 90 - 58 = 32$.