AMC10 2005 A

AMC10 2005 A · Q16

AMC10 2005 A · Q16. It mainly tests Remainders & modular arithmetic, Digit properties (sum of digits, divisibility tests).

The sum of the digits of a two-digit number is subtracted from the number. The units digit of the result is 6. How many two-digit numbers have this property?

一个两位数的各位数字之和从该数中减去，结果的个位数是6。有多少个两位数具有这个性质？

(A) 5 5

(B) 7 7

(D) 10 10

(E) 19 19

Answer

Correct choice: (D)

正确答案：（D）

Solution

Let $10a+b$ be the two-digit number. When $a+b$ is subtracted the result is $9a$. The only two-digit multiple of 9 that ends in 6 is $36$, so $a=4$. The ten numbers between 40 and 49 have this property.

设两位数为 $10a+b$。减去 $a+b$ 后得到 $9a$。以6结尾的两位数9的倍数只有 $36$，所以 $a=4$。40到49之间的十个数具有这个性质。

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