AMC 8 2022 Problem 3
AMC8 2022 年第 3 题
整除与因数★★☆☆☆
When three positive integers , , and are multiplied together, their product is 100. Suppose a < b < c. In how many ways can the numbers be chosen?
当三个正整数a、b和c相乘时，它们的乘积是 100。假设a＜b＜c。问这些数共有多少种选择方法？
- A.0
- B.1
- C.2
- D.3
- E.4
正确答案 · Correct Answer
E
解析 · Solution
Solution 1
The positive divisors of 100 are $1,2,4,5,10,20,25,50,100.$ It is clear that so we apply casework to c: If c=10, then (a,b,c)=(2,5,10). If c=20, then (a,b,c)=(1,5,20). If c=25, then (a,b,c)=(1,4,25). If c=50, then (a,b,c)=(1,2,50). Together, the numbers $a,b,$ and can be chosen in 4 ways.
The positive divisors of 100 are $1,2,4,5,10,20,25,50,100.$ We apply casework to : If a=1, then there are 3 cases: b=2,c=50b=4,c=25b=5,c=20 If a=2, then there is only 1 case: b=5,c=10 In total, there are 3+1=4 ways to choose distinct positive integer values of a,b,c.
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