2010 AMC 8 Problems/Problem 22
Problem
The hundreds digit of a three-digit number is 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?
Solution 1
Let the hundreds, tens, and units digits of the original three-digit number be , , and , respectively. We are given that . The original three-digit number is equal to . The hundreds, tens, and units digits of the reversed three-digit number are , , and , respectively. This number is equal to . Subtracting this expression from the expression for the original number, we get . Thus, the units digit in the final result is
Solution 2
The result must hold for any three-digit number with hundreds digit being more than the units digit. is such a number. Evaluating, we get . Thus, the units digit in the final result is
Solution 3
Set the units digit of the original number as . Thus, its hundreds digit is . After the digits are reversed, the hundreds digit of the original number will be the units digit of the new number. Since , we can do regrouping and "borrow" from the tens digit and bring it to the units digit as a . Therefore, the units digit will end up as
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See Also
2010 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
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