AMC 8 2013 Problem 10

AMC8 2013 年第 10 题

整除与因数★★☆☆☆

What is the ratio of the least common multiple of 180 and 594 to the greatest common factor of 180 and 594?

180 和 594 的最小公倍数和最大公约数的比值是多少？

A.110
B.165
C.330
D.625
E.660

正确答案 · Correct Answer

解析 · Solution

Solution 1

To find either the LCM or the GCF of two numbers, always prime factorize first. The prime factorization of

180 = 3^2×5 \times 2^2

. The prime factorization of

594 = 3^3×11×2

. Then, to find the LCM, we have to find the greatest power of all the numbers there are; if one number is one but not the other, use it (this is

3^3, 5, 11, 2^2

). Multiply all of these to get 5940. For the GCF of 180 and 594, use the least power of all of the numbers that are in both factorizations and multiply.

3^2×2

= 18. Thus the answer =

\displaystyle \frac{5940}{18}

= 330.

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