AMC10 2001 A

AMC10 2001 A · Q6

AMC10 2001 A · Q6. 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 $n$ 的各位数字的乘积和之和。例如，$P(23) = 6$ 和 $S(23) = 5$。假设 $N$ 是一个两位数，使得 $N = P(N) + S(N)$。$N$ 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

Suppose $N = 10a + b$. Then $10a + b = ab + (a+b)$. It follows that $9a = ab$, which implies that $b = 9$, since $a \neq 0$.

设 $N = 10a + b$。则 $10a + b = ab + (a+b)$。由此得出 $9a = ab$，这意味着 $b = 9$，因为 $a \neq 0$。

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