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Difference between revisions of "2012 AMC 10B Problems/Problem 9"

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== Solution ==
 
== Solution ==
  
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Out of the first two integers, it's possible for both to be odd: for example, <math>11 + 15 = 26.</math> But the next two integers, when added, increase the sum by <math>15,</math> which is odd, so one of them must be odd and the other must be even: for example, <math>3 + 12 = 15.</math> Finally, the next two integers increase the sum by <math>16,</math> which is even, so we can have both be odd: for example, <math>9 + 7 = 16.</math> Therefore, <math>\boxed{\textbf{(A) } 1}</math> is the minimum number of integers that must be even.
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Out of the first two integers, it's possible for both to be even: for example, <math>10 + 16 = 26.</math> But the next two integers, when added, increase the sum by <math>15,</math> which is odd, so one of them must be odd and the other must be even: for example, <math>3 + 12 = 15.</math> Finally, the next two integers increase the sum by <math>16,</math> which is even, so we can have both be even: for example, <math>2 + 14 = 16.</math> Therefore, <math>\boxed{\textbf{(A) } 1}</math> is the minimum number of integers that must be odd.
  
  
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Sidenote by Williamgolly:
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You can use the sum of each pairs of numbers and take mod 2. See [[modular arithmetic]]
  
 
==See Also==
 
==See Also==

Latest revision as of 16:46, 5 October 2022

Problem 9

Two integers have a sum of 26. When two more integers are added to the first two integers the sum is 41. Finally when two more integers are added to the sum of the previous four integers the sum is 57. What is the minimum number of odd integers among the 6 integers?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

Solution

Out of the first two integers, it's possible for both to be even: for example, $10 + 16 = 26.$ But the next two integers, when added, increase the sum by $15,$ which is odd, so one of them must be odd and the other must be even: for example, $3 + 12 = 15.$ Finally, the next two integers increase the sum by $16,$ which is even, so we can have both be even: for example, $2 + 14 = 16.$ Therefore, $\boxed{\textbf{(A) } 1}$ is the minimum number of integers that must be odd.


Sidenote by Williamgolly: You can use the sum of each pairs of numbers and take mod 2. See modular arithmetic

See Also

2012 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
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 10 Problems and Solutions

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