AMC8 2006

AMC8 2006 · Q11

AMC8 2006 · Q11. It mainly tests Perfect squares & cubes, Digit properties (sum of digits, divisibility tests).

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

有多少个两位数的各位数字之和是完全平方数？

(A) 13 13

(B) 16 16

(D) 18 18

(E) 19 19

Answer

Correct choice: (C)

正确答案：（C）

Solution

(C) The sum of the digits of a two-digit number is at most 9 + 9 = 18. This means the only possible perfect square sums are 1, 4, 9 and 16. Each square has the following two-digit possibilities: 1 : 10 4 : 40, 31, 22, 13 9 : 90, 81, 72, 63, 54, 45, 36, 27, 18 16 : 97, 88, 79 There are 17 two-digit numbers in all.

(C) 两位数的各位数字之和最大为 $9+9=18$。因此，可能的完全平方和只有 $1,4,9,16$。每个平方对应的两位数情况如下： 1：10 4：40，31，22，13 9：90，81，72，63，54，45，36，27，18 16：97，88，79 总共有 17 个两位数。

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