AMC12 2001 A

AMC12 2001 A · Q2

AMC12 2001 A · Q2. It mainly tests Linear equations, Digit properties (sum of digits, divisibility tests).

Let $P(n)$ and $S(n)$ denote the product and the sum, respectively, of the digits of the integer $n$. For example, $P(23) = 6$ and $S(23) = 5$. Suppose $N$ is a two-digit number such that $N = P(N)+S(N)$. What is the units digit of $N$?

设 $P(n)$ 和 $S(n)$ 分别表示整数 $n$ 的各位数字的积与和。例如，$P(23) = 6$ 且 $S(23) = 5$。假设 $N$ 是一个两位数，满足 $N = P(N)+S(N)$。$N$ 的个位数字是多少？

(A) 2 2

(B) 3 3

(D) 8 8

(E) 9 9

Answer

Correct choice: (E)

正确答案：（E）

Solution

Denote $a$ and $b$ as the tens and units digit of $N$, respectively. Then $N = 10a+b$. It follows that $10a+b=ab+a+b$, which implies that $9a=ab$. Since $a\neq0$, $b=9$. So the units digit of $N$ is $\boxed{\textbf{(E) }9}$.

设 $a$ 和 $b$ 分别为 $N$ 的十位数字和个位数字，则 $N = 10a+b$。于是 $10a+b=ab+a+b$，从而 $9a=ab$。由于 $a\neq0$，所以 $b=9$。因此 $N$ 的个位数字是 $\boxed{\textbf{(E) }9}$。

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