Problem 10 · 1998 AJHSME
Hard
Number Theory
casework
Each of the letters W, X, Y, and Z represents a different integer in the set {1, 2, 3, 4}, but not necessarily in that order. If WX − YZ = 1, then the sum of W and Y is
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Answer: E — 7.
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Hint 1 of 2
The difference comes out to exactly 1 — a clean whole number with no leftover fraction. That's a big clue: the easiest way to get a whole number is for each fraction to already BE a whole number.
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Hint 2 of 2
From {1, 2, 3, 4}, which fractions are whole numbers? You need a denominator that divides its numerator. Only a 1 or a 2 on the bottom can do that here.
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Approach: force both fractions to be whole numbers
- The result is the whole number 1, so chase the cleanest setup: make each fraction a whole number. From {1,2,3,4}, a fraction is whole only with bottom 1 (any top) or bottom 2 under top 4. To use both 1 and 2 as bottoms once each, the fractions must be 4/2 = 2 and (3 or odd)/1.
- Take W/X = 3/1 = 3 and Y/Z = 4/2 = 2: then 3 − 2 = 1. Every letter is a different number from {1,2,3,4}, as required.
- So W = 3 and Y = 4, giving W + Y = 7 — which is the largest choice, a nice confirmation we found the intended setup.
- Why this transfers: when a messy expression must equal a clean whole number, look for the structure that forces it cleanly (here, integer fractions) instead of testing every arrangement at random.
Mark:
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