, 2025

In a list of numbers having exactly one mode, the mode is the
number that appears in the list the greatest number of times. Since the
mode of the given list is $8$, and
the list has exactly one mode, then $8$ must appear more times than each of the
other numbers in the list.

If $n=8$, then $8$ appears in the list three times, which
is more than any other number in the list, and so the list has exactly
one mode, which is $8$, as
required.

If the value of $n$ is $15$ or $3$, then the list has three numbers that
each appear twice, and thus the list does not have exactly one
mode.

If $n=9$, then $9$ appears more times than any other
number and so the mode is $9$.

If $n=10$, then both $8$ and $9$ each appear twice (which is more than
any other number in the list), and thus the list does not have exactly
one mode.