AMC8 2013

AMC8 2013 · Q10

AMC8 2013 · Q10. It mainly tests Primes & prime factorization, GCD & LCM.

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 的最小公倍数与最大公因数的比值是多少？

(A) 110 110

(B) 165 165

(D) 625 625

(E) 660 660

Answer

Correct choice: (C)

正确答案：（C）

Solution

To find either the LCM or the GCF of two numbers, always prime factorize first. The prime factorization of $180 = 3^2 \times 5 \times 2^2$. The prime factorization of $594 = 3^3 \times 11 \times 2$. 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. $3^2 \times 2$ = 18. Thus the answer = $\frac{5940}{18}$ = $\boxed{\textbf{(C)}\ 330}$.

求两个数的 LCM 或 GCF 时，总是先进行质因数分解。 $180 = 3^2 \times 5 \times 2^2$。 $594 = 3^3 \times 11 \times 2$。 LCM 取所有质因数的最大幂：$3^3, 5, 11, 2^2$，相乘得 5940。 GCF 取共同质因数的最小幂：$3^2 \times 2 = 18$。因此答案$= \frac{5940}{18} = \boxed{\textbf{(C)}\ 330}$。

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