AMC10 2013 B

AMC10 2013 B · Q9

AMC10 2013 B · Q9. It mainly tests Primes & prime factorization, GCD & LCM.

Three positive integers are each greater than 1, have a product of 27,000, and are pairwise relatively prime. What is the sum of these integers?

三个大于1的正整数，积为27,000，且两两互质。这些整数的和是多少？

(A) 100 100

(B) 137 137

(D) 160 160

(E) 165 165

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): Note that $27{,}000 = 2^3 \cdot 3^3 \cdot 5^3.$ The only three pairwise relatively prime positive integers greater than 1 with a product of 27,000 are 8, 27, and 125. The sum of these numbers is 160.

答案（D）：注意 $27{,}000 = 2^3 \cdot 3^3 \cdot 5^3.$ 乘积为 27,000 且两两互质并且大于 1 的三个正整数只有 8、27 和 125。它们的和为 160。

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