Six positive integers have a mean of 15, a median of 15 and a mode of 16.
The range of these numbers is 7. Find the six numbers. Answer ... ... ... ... ... ...

1. Use the mean to find the total, and use the median to restrict the middle two numbers.

Arrange the six positive integers in ascending order:

$a \leq b \leq c \leq d \leq e \leq f$.

Mean $=15$, so

$a+b+c+d+e+f=6\times 15=90.$

For six numbers, the median is the average of the 3rd and 4th numbers:

$\frac{c+d}{2}=15$

$c+d=30.$

2. Use the mode and range to determine the six integers.

The mode is $16$, so $16$ must occur more often than any other number.

Since $c+d=30$ and the data are ordered, the middle pair must allow $16$ to appear as the mode. The possible middle pair is

$c=14,\quad d=16.$

So the numbers are

$a,\ b,\ 14,\ 16,\ e,\ f.$

The range is $7$, so

$f-a=7.$

Also,

$a+b+14+16+e+f=90$

$a+b+e+f=60.$

Trying values consistent with the order and range gives

$a=12,\quad f=19.$

Then

$12+b+e+19=60$

$b+e=29.$

To make $16$ the mode, take $e=16$, so

$b=13.$

Hence the six numbers are

$12,\ 13,\ 14,\ 16,\ 16,\ 19.$