## Thursday, October 16, 2014

### 282: Number puzzle

"The sum of two positive integers is 216. The greatest common factor of the two numbers is 24. What are all the possible pairs of numbers?"

What approach seems the easiest here?
I can see using solution methods such as:
• algebra
• guess and check
• organized list
• visual representation
Which (or which other) method rings true for you?
Which of the two sentences eliminates the most numbers?

1. Which approach seems the easiest here?
Excel
Column A Column B Column C
24 | =216-a1 | =gcd(a1,b1)
=a1+24

Drag the formulas and eliminate wrong answers
24 192 24
48 168 24
72 144 72
96 120 24
120 96 24
144 72 72
168 48 24
192 24 24

2. Let 24x represent the first number and 24y represent the second number, where x and y are relative prime positive integers and x < y. Thus, 24x + 24y = 216 => x + y = 9. The only pairs of values (x, y) that work for that equation are (1, 8), (2, 7), and (4, 5).

Therefore, the three pairs of numbers are (24, 192), (48, 168), and (96, 120).

1. Relatively prime, not relative prime.