Five positive integers have a mean of 10 and a median of 12, and their only mode is 15. What is the greatest possible range of the five integers?

Sorted: $a<b<12$ ≤ 15 = 15 (the mode must appear twice above the median; three 15s would make the median 15). Then $a+b=50-12-30=8$ with a ≠ $b$. The range $15-a$ is largest when a is smallest: $a=1,b=7$, range 14.