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<b<c. 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%

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About mvtrinh

Retired high school math teacher.
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