Question

The mean of six positive integers is 15. The median is 18, and the only mode of the integers is less than 18. The maximum possible value of the largest of the six integers is

Accepted Answer

D. Sum of 6 integers = $15 	imes 6 = 90$. 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 $x_3 
eq x_4$ (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 $x_6 = 90 - (1 + 1 + 17 + 19 + 20) = 90 - 58 = 32$.

Mean, Median and Mode of Integers

Explanation

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