Difference between revisions of "2014 AMC 8 Problems/Problem 8"
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==Problem== | ==Problem== | ||
| − | Eleven members of the Middle School Math Club each paid the same amount for a guest speaker to talk about problem solving at their math club meeting. They paid their guest speaker <math> \underline{1} \underline{A} \underline{2}</math>. What is the missing digit <math>A</math> of this <math>3</math>-digit number? | + | Eleven members of the Middle School Math Club each paid the same amount for a guest speaker to talk about problem solving at their math club meeting. They paid their guest speaker <math> \textdollar\underline{1} \underline{A} \underline{2}</math>. What is the missing digit <math>A</math> of this <math>3</math>-digit number? |
<math>\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad \textbf{(E) }4</math> | <math>\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad \textbf{(E) }4</math> | ||
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==Solution== | ==Solution== | ||
A number is divisible by 11 iff the difference between the sum of the digits in the odd-numbered slots (e.g. the ones slot, the hundreds slot, etc.) and the sum in the even-numbered slots (e.g. the tens slot, the thousands slot) is a multiple of 11. So 1 + 2 - A is equivalent to 0 mod 11. Clearly 3 - A cannot be equal to 11 or any multiple of 11 greater than that. So 3 - A = 0 --> A = 3. | A number is divisible by 11 iff the difference between the sum of the digits in the odd-numbered slots (e.g. the ones slot, the hundreds slot, etc.) and the sum in the even-numbered slots (e.g. the tens slot, the thousands slot) is a multiple of 11. So 1 + 2 - A is equivalent to 0 mod 11. Clearly 3 - A cannot be equal to 11 or any multiple of 11 greater than that. So 3 - A = 0 --> A = 3. | ||
Revision as of 08:50, 27 November 2014
Problem
Eleven members of the Middle School Math Club each paid the same amount for a guest speaker to talk about problem solving at their math club meeting. They paid their guest speaker 
. What is the missing digit 
of this 
-digit number?
Solution
A number is divisible by 11 iff the difference between the sum of the digits in the odd-numbered slots (e.g. the ones slot, the hundreds slot, etc.) and the sum in the even-numbered slots (e.g. the tens slot, the thousands slot) is a multiple of 11. So 1 + 2 - A is equivalent to 0 mod 11. Clearly 3 - A cannot be equal to 11 or any multiple of 11 greater than that. So 3 - A = 0 --> A = 3.
See Also
| 2014 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
| 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 AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
