Problem 12 · 2020 Math Kangaroo
Stretch
Algebra & Patterns
substitutionfactorization
Integers a, b, c and d satisfy the equality \(2ab = 3cd\). Which of the following numbers can be the product abcd?
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Answer: C — 150
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Hint 1 of 2
Treat ab and cd as single quantities and use the relation to write abcd with just one of them.
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Hint 2 of 2
You’ll find abcd must be 6 times a perfect square.
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Approach: substitute cd = 2ab/3 to express the product
- From 2ab = 3cd we get cd = 2ab/3, so abcd = ab·cd = 2(ab)²/3.
- For an integer, ab must be a multiple of 3, giving abcd = 6k².
- Among the options only 150 = 6·5² has this form.
- So the product can be 150.
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