2015 AMC 8 Problems/Problem 14
Contents
Problem
Which of the following integers cannot be written as the sum of four consecutive odd integers?
Solutions
Solution 1
Let our numbers be , where is odd. Then, our sum is . The only answer choice that cannot be written as , where is odd, is .
Solution 2
If the four consecutive odd integers are and ; then, the sum is . All the integers are divisible by except .
Solution 3
If the four consecutive odd integers are and , the sum is , and divided by gives . This means that must be even. The only integer that does not give an even integer when divided by is , so the answer is .
Solution 4
From Solution 1, we have the sum of the numbers to be equal to . Taking mod 8 gives us for some residue and for some odd integer . Since , we can express it as the equation for some integer . Multiplying 4 to each side of the equation yields , and taking mod 8 gets us , so . All the answer choices except choice D is a multiple of 8, and since 100 satisfies all the conditions the answer is .
Solution 5
Since they want CONSECUTIVE odd numbers, it won't be hard to just list the sums out: All of the answer choices can be a sum of consecutive odd numbers except , so the answer is .
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See Also
2015 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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