2015 AMC 8 Problems/Problem 14
Which of the following integers cannot be written as the sum of four consecutive odd integers?
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
As in 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
.
See Also
2015 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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