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If x is a sum of all even integers on the interval 13...65 a

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by Scott@TargetTestPrep » Sat Jul 06, 2019 5:00 pm
BTGmoderatorDC wrote:If x is a sum of all even integers on the interval 13...65 and y is their number, what is the GCD (x, y)?

(A) 1
(B) 13
(C) 26
(D) 52
(E) 1014

OA C

Source: Princeton Review
The number of even integers in this evenly-spaced set of numbers from 14 to 64, inclusive, is:

(64 - 14)/2 + 1 = 26 = y

The average of the numbers in an evenly-spaced set is calculated as: (largest number + smallest number) / 2. Thus, the sum of these 26 integers is:

(64 + 14)/2 * 26 = 39 * 26 = x

Thus, the GCF(x, y) is 26.

Answer: C

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by swerve » Sat Jul 06, 2019 6:19 pm
\(y\) is the number of even integers from 14 to 64, inclusive = (largest - smallest)/2 + 1 = (64 - 14)/2 + 1 = 26.

\(x = 14 + 16 + ... + 64 =\) (largest + smallest)/2 *(# of terms) = \((14 + 64)/2\cdot26 = 39\cdot26\).

GCD of 26 and 39\(\cdot\)26 is 26.

Therefore, the correct answer is __C__
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