AMC 8 2001 Problem 2
AMC8 2001 年第 2 题
方程与不等式★☆☆☆☆
I'm thinking of two whole numbers. Their product is 24 and their sum is 11. What is the larger number?
我在想两个整数。他们的积是 24，和是 11。较大的那个数是多少？
- A.3
- B.4
- C.6
- D.8
- E.12
正确答案 · Correct Answer
D
解析 · Solution
Solution 1
Let the numbers be and . Then we have x+y=11 and xy=24. Solving for in the first equation yields x=11-y, and substituting this into the second equation gives (11-y)(y)=24. Simplifying this gives , or . This factors as (y-3)(y-8)=0, so y=3 or y=8, and the corresponding values are x=8 and x=3. These are essentially the same answer: one number is 3 and one number is 8, so the largest number is .
Use the answers to attempt to "reverse engineer" an appropriate pair of numbers. Looking at option , guess that one of the numbers is 3. If the sum of two numbers is 11 and one is 3, then other must be 11 - 3 = 8. The product of those numbers is , which is the second condition of the problem, so our numbers are 3 and 8. However, 3 is the smaller of the two numbers, so the answer is 8 or .
We go through the divisor pairs of 24 to see which two numbers sum to 11. The numbers clearly cannot be negative. If one was negative, then the other must also be negative in order to multiply to a positive product, but it would be impossible for the numbers to add up to a positive sum. So, we look at the positive divisor pairs of 24, namely 1 and 24, 2 and 12, 3 and 8, and 4 and 6. The only pair of numbers that sums to 11 is 3 and 8. The larger number is 8, so the answer is .
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