Five positive integers have a mean of 5, median of 4, mode of 3 and a range of 7. Find the five numbers. Answer ___

1. Use the median and mode to fix the first three numbers in order.

Let the five positive integers in ascending order be
$a \leq b \leq c \leq d \leq e.$
Since the median is $4$, the middle number is
$c=4.$
Since the mode is $3$, the number $3$ must occur most often. Hence the two numbers before $4$ are
$a=3,\quad b=3.$

2. Use the range and mean to find the remaining two numbers.

The range is $7$, so
$e-a=7.$
Since $a=3$,
$e=10.$
The mean is $5$, so the total of the five numbers is
$5\times 5=25.$
Thus
$3+3+4+d+10=25,$
$d=5.$