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 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
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