Problem 9 · 2023 Math Kangaroo
Medium
Number Theory
factor-pairs
How many pairs of positive integers \((a, b)\) fulfil the equation \(\frac{a}{5} = \frac{7}{b}\)?
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Answer: E — 4
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Hint 1 of 2
Cross-multiply to turn the equation into a product.
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Hint 2 of 2
Count the factorizations of that product into two positive integers.
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Approach: cross-multiply and count factor pairs
- a/5 = 7/b means a·b = 35.
- The positive factor pairs of 35 are (1,35), (5,7), (7,5), (35,1).
- Each gives a valid (a, b).
- So there are 4 pairs.
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