AMC 8 1994 Problem 10
AMC8 1994 年第 10 题
整除与因数★★☆☆☆
For how many positive integer values of is the expression 36/N+2 an integer?
有多少个正整数值的 能使表达式 为整数？
- A.7
- B.8
- C.9
- D.10
- E.12
正确答案 · Correct Answer
A
解析 · Solution
Solution 1
We should list all the positive divisors of 36 and count them. By trial and error, the divisors of 36 are found to be 1,2,3,4,6,9,12,18,36, for a total of 9. However, 1 and 2 can't be equal to N+2 for a POSITIVE integer N, so the number of possibilities is .
To find the number of positive divisors of 36, first prime factorize 36 to get . Then add 1 to the power of both 2 and 3 to get 3. Multiply 3*3 to get 9. Since the problem is asking only for positive integer values of N, subtract 2 from 9 (since 2-2 and 1-2 result in integers that are not positive) to get .
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