AMC 8 2008 Problem 22
AMC8 2008 年第 22 题
整除与因数★★★☆☆
For how many positive integer values of n are both and 3n three-digit whole numbers?
有多少个正整数 ，使得 和 都是三位整数？
- A.12
- B.21
- C.27
- D.33
- E.34
正确答案 · Correct Answer
A
解析 · Solution
Solution 1
Instead of finding , we find . We want and to be three-digit whole numbers. The smallest three-digit whole number is 100, so that is our minimum value for , since if , then . The largest three-digit whole number divisible by 9 is 999, so our maximum value for is . There are 12 whole numbers in the closed set , so the answer is 12. - ColtsFan10
We can set the following inequalities up to satisfy the conditions given by the question: , and . Once we simplify these and combine the restrictions, we get the inequality . Now we have to find all multiples of 3 in this range for to be an integer. We can compute this by setting , where . Substituting for in the previous inequality, we get , and there are 111-100+1 integers in this range, giving us the answer, 12.
We can create a list of the positive integers that fulfill the requirement of and being three-digit whole numbers. The first number of this list must be 300 since is the smallest positive integer that satisfies this requirement. The last number of this list must be 333 since is the largest positive integer that satisfies this requirement. Since the problem requires and to be whole numbers, the other numbers must be multiples of 3 (just like 300 and 333), so the list would look like this: To put this list into a countable form we must put it in a form similar to . So, we manipulate it as follows: Now we can tell that there are 12 positive integers which satisfy the two requirements, so the answer is 12.
We have to find the range for all numbers first. We can first find the smallest digit in which is a 3 digit number that 3n is a 3 digit number which is = 100, then we find the biggest number in which is a 3 digit number and 3n is a 3 digit number which is = 111. Now we know that all integers from 100 to 111 are possible values and the amount of integer values in that range is 12 (note I divided the numbers by 3 for convenience)
继续练习