AMC 8 1988 Problem 14
AMC8 1988 年第 14 题
整除与因数★★★☆☆
and are whole numbers and . The largest possible value of is
和 是正整数，且 。 的最大可能值是
- A.12
- B.13
- C.15
- D.20
- E.37
正确答案 · Correct Answer
E
解析 · Solution
Solution 1
Since and can be swapped to get the same sum, only cases where need to be considered. These cases are shown in the table:
$\begin{array}{|c|c|c|} \hline \diamondsuit & \Delta & \diamondsuit + \Delta \\ \hline 36 & 1 & 37 \\ \hline 18 & 2 & 20 \\ \hline 12 & 3 & 15 \\ \hline 9 & 4 & 13 \\ \hline 6 & 6 & 12 \\ \hline \end{array}
$ So 37 is the largest possible sum.We notice that the function is convex with a minimum at x = 6, so we seek to pick a value of far from 6. To ensure that the answers are positive integers, we let x = 36, from which we find that f(x) = 37, the maximum.
-GaloisTorrent
Notice that 1×36 = 36 and 1+36=37. Since 37 is the highest answer choice it must be correct.
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