Problem 9 · 1987 AJHSME
Medium
Number Theory
lcm-from-prime-factors
When finding the sum 1⁄2 + 1⁄3 + 1⁄4 + 1⁄5 + 1⁄6 + 1⁄7, the least common denominator used is
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Answer: C — 420.
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Hint 1 of 2
The common denominator must be a multiple of every bottom — but it doesn't need to be their product. What's the smallest number all of 2–7 divide into?
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Hint 2 of 2
Build the LCM from primes: include each prime to the highest power that shows up among 2, 3, 4, 5, 6, 7. The 6 brings nothing new (it's just 2 × 3, already covered).
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Approach: take the highest power of each prime
- Factor the denominators: 2, 3, 2², 5, 2·3, 7. The distinct prime powers needed are 2² (from the 4), 3, 5, and 7 — note 6 = 2 × 3 is already accounted for.
- LCM = 2² × 3 × 5 × 7 = 4 × 3 × 5 × 7 = 420.
- Why this transfers: the least common denominator is the LCM, NOT the product. Multiplying all six gives 5040 (the trap answer) — far bigger than needed, because it double-counts shared factors like the 2 in 4 and 6.
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