Question

一个两位正整数称为 可爱数 ，如果它等于其非零十位数字与个位数字平方之和。共有多少个两位正整数是可爱数？

A two-digit positive integer is said to be cuddly if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly?

$0$

$1$

$2$

$3$

$4$

答案：B

知识点：数字丢番图方程整除性

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解答：

设 $\underline{a} \ \underline{b}$ 是一个 $2$ 位可爱数。

由定义，化简得因此 $b-1$ 必须整除 $b$ 与 $3$ 的乘积。相邻整数中至多一个能被 $3$ 整除，所以或必须能被九整除。 $10a + b = a + b^2.$ $9a = b(b - 1).$

因此 $b$ 只能是 $b=0,1$ 或 $9$ 。若，则 $a=0$ ，不能组成两位数；若 $b=9$ ，则 $a=8$ 。检验可知 $89$ 是可爱数，因此只有 $1$ 个。

所以正确答案是 B。

Let $\underline{a} \ \underline{b}$ be a $2$ -digit cuddly number.

Then $10a + b = a + b^2.$ Rearranging, we get $9a = b(b - 1).$ Because consecutive integers cannot both be divisible by $3,$ one of $b$ and $b-1$ must contain both factors of $3.$

For a digit $b,$ this leaves $b=0,1,$ or $9.$ The first two choices give $a=0,$ which does not make a two-digit number. If $b=9,$ then $a=8.$ Checking, we get that $89$ is a cuddly number. This shows that there is only $1$ two-digit cuddly number.

Thus, B is the correct answer.

2021 AMC 10A Fall 第 8 题

答案：B

难度评级：1210

解答：

其他年份的第 8 题