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If x is the sum of the first 10 positive multiples of 3 and

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by Max@Math Revolution » Thu May 03, 2018 11:36 pm

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[GMAT math practice question]

If x is the sum of the first 10 positive multiples of 3 and y is the sum of the first 10 positive multiples of 2, which of the following is a factor of x - y?

A. 20
B. 35
C. 55
D. 75
E. 90
Last edited by Max@Math Revolution on Sun May 06, 2018 6:08 am, edited 1 time in total.
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by GMATGuruNY » Fri May 04, 2018 5:05 am
Max@Math Revolution wrote:[GMAT math practice question]

If x is the sum of the first 10 positive multiples of 3 and y is the sum of the first 10 positive multiples of 2, which of the following is a factor of x - y?

A. 2
B. 3
C. 5
D. 7
E. 9
For any EVENLY SPACED SET:
AVERAGE = (BIGGEST + SMALLEST)/2.
SUM = (COUNT)(AVERAGE).

The first 10 positive multiples of 3 are as follows:
3, 6, 9, 12, 15, 18, 21, 24, 27, 30.
Thus:
Average = (biggest + smallest)/2 = (30+3)/2 = 16.5.
x = sum = (count)(average) = 10*16.5 = 165.

The first 10 positive multiples of 2 are as follows:
2, 4, 6, 8, 10, 12, 14, 16, 18, 20.
Thus:
Average = (biggest + smallest)/2 = (20+2)/2 = 11.
y = sum = (count)(average) = 10*11 = 110.

Result:
x-y = 165 - 110 = 55 = 5*11.
Of the five answer choices, only C is a factor of x-y.

The correct answer is C.
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by Brent@GMATPrepNow » Fri May 04, 2018 5:40 am
Max@Math Revolution wrote: If x is the sum of the first 10 positive multiples of 3 and y is the sum of the first 10 positive multiples of 2, which of the following is a factor of x - y?

A. 2
B. 3
C. 5
D. 7
E. 9
x is the sum of the first 10 positive multiples of 3
So, x = 3 + 6 + 9 + 12 + ....
Factor out the 3 to get: x = 3(1 + 2 + 3 + 4 + . . . . + 9 + 10)

y is the sum of the first 10 positive multiples of 2
So, y = 2 + 4 +6 + 8 + 10 + ...
Factor out the 2 to get: y = 2(1 + 2 + 3 + 4 + . . . . + 9 + 10)

Which of the following is a factor of x - y?
x - y = 3(1 + 2 + 3 + 4 + . . . . + 9 + 10) - 2(1 + 2 + 3 + 4 + . . . . + 9 + 10)
= 1(1 + 2 + 3 + 4 + . . . . + 9 + 10)
= 1 + 2 + 3 + 4 + . . . . + 9 + 10
= 55

Check the answer choices....
Only answer choice C (5) is a factor of 55

Answer: C

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Brent Hanneson - Creator of GMATPrepNow.com
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by Max@Math Revolution » Mon May 07, 2018 1:09 am
=>
x - y = ( 3 + 6 + 9 + ... + 30 ) - ( 2 + 4 + 6 + ... + 20 )
= 3(1 + 2 + ... + 10 ) - 2( 1 + 2 + 3 + ... + 10 )
= ( 3 - 2 ) ( 1 + 2 + 3 + ... + 10 )
= 55
= 5 * 11

Therefore, the answer is C.
Answer: C
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by Jeff@TargetTestPrep » Mon May 07, 2018 10:10 am
Max@Math Revolution wrote:[GMAT math practice question]

If x is the sum of the first 10 positive multiples of 3 and y is the sum of the first 10 positive multiples of 2, which of the following is a factor of x - y?

A. 20
B. 35
C. 55
D. 75
E. 90
Recall that the sum = average x quantity. The first positive multiple of 3 is 3 and the last (10th) multiple of 3 is 30. Thus, the average is (30 + 3)/2 = 33/2, and the sum of the first 10 positive multiples of 3 is:

33/2 x 10 = 165

Similarly, the first positive multiple of 2 is 2 and the last (10th) multiple of 2 is 20. Thus the average is (20 + 2)/2 = 11, and the sum of the first 10 positive multiples of 2 is:

11 x 10 = 110

x - y = 165 - 110 = 55 = 5 x 11

We see 5 is a factor of 55.

Alternate Solution:

The first positive multiple of 3 is 3 and the last (10th) multiple of 3 is 30. Thus,

x = 3 + 6 + 9 + ... + 30

Similarly, the first positive multiple of 2 is 2 and the last (10th) multiple of 3 is 20. Thus,

y = 2 + 4 + 6 + ... + 20

Let's subtract the second equation from the first:

x - y = (3 - 2) + (6 - 4) + (9 - 6) + ... + (30 - 20)

x - y = 1 + 2 + 3 + ... + 10

Recall that the formula for the sum of the first n consecutive positive integers is n(n + 1)/2, where n is the largest number in the list. Thus, we have:

x - y = (10 x 11)/2 = 55

We see 5 is a factor of 55.

Answer: C

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