AMC8 1995

AMC8 1995 · Q22

AMC8 1995 · Q22. It mainly tests Primes & prime factorization, Number theory misc.

The number 6545 can be written as a product of a pair of positive two-digit numbers. What is the sum of this pair of numbers?

数字 6545 可以写成一对正的双位数的乘积。这对数的和是多少？

(A) 162 162

(B) 172 172

(D) 174 174

(E) 222 222

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): The prime factorization of 6545 is $5 \times 7 \times 11 \times 17$. Since the product of any three of these primes is a three-digit number and since $7 \times 17$ and $11 \times 17$ are both three-digit numbers, it follows that the only pair of two-digit numbers with product 6545 is $5 \times 17 = 85$ and $7 \times 11 = 77$. Thus the answer is $85 + 77 = 162$.

答案（A）：6545 的质因数分解是 $5 \times 7 \times 11 \times 17$。由于这四个质数中任意三个的乘积都是三位数，并且 $7 \times 17$ 和 $11 \times 17$ 也都是三位数，所以可知乘积为 6545 的两位数乘积组合只有 $5 \times 17 = 85$ 和 $7 \times 11 = 77$。因此答案为 $85 + 77 = 162$。

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