AMC 8 2010 Problem 22
AMC8 2010 年第 22 题
整除与因数★★★☆☆
The hundreds digit of a three-digit number is 2 more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result?
一个三位数的百位比个位大 2。把这个三位数的各个位上数字颠倒，然后把原来的三位数减去这个数，所得最终结果的个位数字是多少？
- A.0
- B.2
- C.4
- D.6
- E.8
正确答案 · Correct Answer
E
解析 · Solution
Solution 1
Let the hundreds, tens, and units digits of the original three-digit number be , , and , respectively. We are given that a=c+2. The original three-digit number is equal to 100a+10b+c = 100(c+2)+10b+c = 101c+10b+200. The hundreds, tens, and units digits of the reversed three-digit number are , , and , respectively. This number is equal to 100c+10b+a = 100c+10b+(c+2) = 101c+10b+2. Subtracting this expression from the expression for the original number, we get (101c+10b+200) - (101c+10b+2) = 198. Thus, the units digit in the final result is 8
The result must hold for any three-digit number with hundreds digit being 2 more than the units digit. 301 is such a number. Evaluating, we get 301-103=198. Thus, the units digit in the final result is 8
Set the units digit of the original number as . Thus, its hundreds digit is x+2. After the digits are reversed, the hundreds digit of the original number will be the units digit of the new number. Since x-(x+2) = -2, we can do regrouping and "borrow" 1 from the tens digit and bring it to the units digit as a 10. Therefore, the units digit will end up as -2 + 10 = 8 Video by MathTalks https:
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