AMC8 2014

AMC8 2014 · Q22

AMC8 2014 · Q22. It mainly tests Linear equations, Factoring.

A $2$-digit number is such that the product of the digits plus the sum of the digits is equal to the number. What is the units digit of the number?

一个两位数满足：数字的乘积加上数字的和等于该数。这个数的个位数字是多少？

(A) 1 1

(B) 3 3

(D) 7 7

(E) 9 9

Answer

Correct choice: (E)

正确答案：（E）

Solution

We can think of the number as $10a+b$, where a is the tens digit and b is the unit digit. Since the number is equal to the product of the digits ($ab$) plus the sum of the digits ($a+b$), we can say that $10a+b=ab+a+b$. We can simplify this to $10a=ab+a$, which factors to $(10)a=(b+1)a$. Dividing by $a$, we have that $b+1=10$. Therefore, the units digit, $b$, is $\boxed{\textbf{(E) }9}$

我们可以将这个数表示为 $10a+b$，其中 $a$ 是十位数字，$b$ 是个位数字。由于这个数等于数字的乘积 ($ab$) 加上数字的和 ($a+b$)，所以有 $10a+b=ab+a+b$。我们可以将其简化为 $10a=ab+a$，可因式分解为 $(10)a=(b+1)a$。两边除以 $a$，得到 $b+1=10$。因此，个位数字 $b$ 是 $\boxed{\textbf{(E) }9}$。

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