AMC 8 2006 Problem 11

AMC8 2006 年第 11 题

整除与因数★★★☆☆

How many two-digit numbers have digits whose sum is a perfect square?

有多少个两位数，满足各个位上数字之和是个完全平方数？

A.13
B.16
C.17
D.18
E.19

正确答案 · Correct Answer

解析 · Solution

Solution 1

There is 1 integer whose digits sum to 1: 10. There are 4 integers whose digits sum to 4:

13, 22, 31, \text{and } 40

. There are 9 integers whose digits sum to 9:

18, 27, 36, 45, 54, 63, 72, 81, \text{and } 90

. There are 3 integers whose digits sum to 16:

79, 88, \text{and } 97

. Two digits cannot sum to 25 or any greater square since the greatest sum of digits of a two-digit number is 9 + 9 = 18. Thus, the answer is 1 + 4 + 9 + 3 = 17.

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