2014 AMC 10B Problems/Problem 10
Contents
Problem
In the addition shown below , , , and are distinct digits. How many different values are possible for ?
Solution
Note from the addition of the last digits that . From the addition of the frontmost digits, cannot have a carry, since the answer is still a five-digit number. Therefore .
Using the second or fourth column, this then implies that , so that and . Note that all of the remaining equalities are now satisfied: and . Thus, if we have any such that then the addition will be satisfied. Since the digits must be distinct, the smallest possible value of is , and the largest possible value is . Any of these values can be obtained by taking . Thus we have that , so the number of possible values is
Video Solution
~savannahsolver
See Also
2014 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
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All AMC 10 Problems and Solutions |
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