The greatest common divisor of positive integers m and n is 6.
The least common multiple of m and n is 126. What is the least
possible value of m + n?

Question

The greatest common divisor of positive integers m and n is 6.
The least common multiple of m and n is 126. What is the least
possible value of m + n?

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Given that the greatest common divisor (gcd) of $ m $ and $ n $ is 6 and the least common multiple (lcm) of $ m $ and $ n $ is 126. ‖ ‖ step 2 ⋮ The product of the gcd and lcm of two numbers $ m $ and $ n $ is equal to the product of the numbers themselves, i.e., $ \text{gcd}(m, n) \times \text{lcm}(m, n) = m \times n $. ‖ ‖ step 3 ⋮ Substituting the given values, we get $ 6 \times 126 = m \times n $. ‖ ‖ step 4 ⋮ Simplifying, we find $ m \times n = 756 $. ‖ ‖ step 5 ⋮ Since $ m $ and $ n $ are both divisible by their gcd, which is 6, we can write $ m = 6k $ and $ n = 6l $, where $ k $ and $ l $ are co-prime (since 6 is the gcd). ‖ ‖ step 6 ⋮ Substituting $ m = 6k $ and $ n = 6l $ into $ m \times n = 756 $, we get $ 6k \times 6l = 756 $. ‖ ‖ step 7 ⋮ Simplifying, we find $ k \times l = 21 $. ‖ ‖ step 8 ⋮ The pairs of positive integers that multiply to 21 are (1, 21) and (3, 7). ‖ ‖ step 9 ⋮ The sum $ m + n $ in terms of $ k $ and $ l $ is $ 6k + 6l $. ‖ ‖ step 10 ⋮ For the pair (1, 21), the sum is $ 6(1) + 6(21) = 6 + 126 = 132 $. ‖ ‖ step 11 ⋮ For the pair (3, 7), the sum is $ 6(3) + 6(7) = 18 + 42 = 60 $. ‖ ‖ step 12 ⋮ The least possible value of $ m + n $ is therefore 60, corresponding to the pair (3, 7). ‖ ∻Answer∻ ⚹ 60 ⚹ ∻Key Concept∻ ⚹Product of gcd and lcm of two numbers⚹ ∻Explanation∻ ⚹The product of the gcd and lcm of two numbers is equal to the product of the numbers themselves. By finding the prime factorization of the product, we can determine the smallest possible sum of the two numbers.⚹◊What is the relationship between the greatest common divisor (gcd) and the least common multiple (lcm) of two numbers m and n, given that gcd(m, n) = 6 and lcm(m, n) = 126?⍭ Generate me a similar question◊

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