"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 of the two sentences eliminates the most numbers?

```
The sum of 2 pos int is 216 and their gcf is 24. Find all possibilities.
```

— David Marain (@dmarain) September 24, 2014

Which approach seems the easiest here?

ReplyDeleteExcel

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

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).

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

Relatively prime, not relative prime.

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