2005 AMC 12B Problems/Problem 15
Contents
Problem
The sum of four two-digit numbers is . None of the eight digits is and no two of them are the same. Which of the following is not included among the eight digits?
Solution 1
can be written as the sum of four two-digit numbers, let's say , , , and . Then . The last digit of is , and won't affect the units digits, so must end with . The smallest value can have is , and the greatest value is . Therefore, must equal or .
Case 1:
The only distinct positive integers that can add up to is . So, ,,, and must include four of the five numbers . We have , or . We can add all of , and try subtracting one number to get to , but none of them work. Therefore, cannot add up to .
Case 2:
Checking all the values for ,,,and each individually may be time-consuming, instead of only having solution like Case 1. We can try a different approach by looking at first. If , , or . That means . We know , so the missing digit is
Solution 2
Alternatively, we know that a number is congruent to the sum of its digits mod 9, so , where is some digit. Clearly, .
See also
2005 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
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