## Sum of Integers

The sum of three positive integers is $50$. Taken in pairs, their differences are $3,4$, and $7$. Find all possibilities for the three numbers.
Source: NCTM Teacher, December 2005

SOLUTION
Suppose the three positive integers are $a,b,c$ such that $a. The pairs of difference are $c-a,c-b$ and $b-a$. It is clear that the largest difference $7$ belongs to $c-a$. However, it is not clear how to distribute the two remaining differences $3$ and $4$. We have two possible cases to examine.
Case 1: $b-a=3$ and $c-b=4$
$c=a+7$
$b=a+3$
$a=a$
——————————
$a+b+c=3a+10$
$50=3a+10$
$3a=40$
$a=13.33$ not an integer
Case 2: $b-a=4$ and $c-b=3$
$c=a+7$
$b=a+4$
$a=a$
—————————-
$a+b+c=3a+11$
$50=3a+11$
$3a=39$
$a=13$
Consequently
$b=13+4=17$
$c=13+7=20$

Answer: $13,17,20%$