Patterns & Reasoning

Depth · one concept from first principles · step 2 of 4

Let the LCM shrink the candidate solutions

The equation 3x-2y=7 has infinitely many natural-number solutions. Among them we want the one where x and y have least common multiple 7, and then we report x+y.
Number systemExpressions
Your answer
How to solve
Strategy Eliminate Possibilities — Listing solutions of 3x-2y=7 never ends, so start from the other condition instead. A number always divides any common multiple of itself, so x and y must both divide 7. That leaves only four pairs to test against the equation.
1STEP 1

Turn the LCM condition into a short list

Each of x and y divides their least common multiple, which is 7. The only natural numbers that divide 7 are 1 and 7.

x ∈ {1, 7}, y ∈ {1, 7}
2STEP 2

Test the four pairs in the equation

Check 3x-2y for every pair. Only (7, 7) gives 7, and its least common multiple really is 7.

3 × 7 - 2 × 7 = 21 - 14 = 7
3STEP 3

Add the two values

With x=7 and y=7, the requested sum follows.

x + y = 7 + 7 = 14
Answer
14
Substitute back: 3 × 7 - 2 × 7 = 7, and the least common multiple of 7 and 7 is 7. Both conditions hold, so x+y=14.
Takeaway

When one condition has endless answers, start from the other condition that allows only a few.

  • Turn the LCM condition into a short list
  • Test the four pairs in the equation
  • Add the two values
Where next?
Another one like thissuggested
12 practice problems
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▶ Practice — 12 problems

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