AMC 8 2014 Problem 22
AMC8 2014 年第 22 题
特殊数与不定方程★★★☆☆
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：各个位上数字之积加上各个位上数字之和，所得结果等于原来的两位数。
那么原来两位数的个位数字是多少？
- A.1
- B.3
- C.5
- D.7
- E.9
正确答案 · Correct Answer
E
解析 · Solution
Solution 1
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 , we have that b+1=10. Therefore, the units digit, , is 9
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