If integer k is equal to the sum of all even multiples of 15 between 295 and 615, what is the greatest prime factor of k?
1)5
2)7
3)11
4)13
5)17
Thank you,
Prerna
Greatest prime factor
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For any EVENLY SPACED SET:prernamalhotra wrote:If integer k is equal to the sum of all even multiples of 15 between 295 and 615, what is the greatest prime factor of k?
1)5
2)7
3)11
4)13
5)17
Thank you,
Prerna
Number of terms = (biggest-smallest)/interval + 1, where the interval is the distance between successive terms.
Average = median = (biggest + smallest)/2.
Sum = (number of terms)(average).
The even multiples of 15 between 295 and 615 are the MULTIPLES OF 30 between 295 and 615.
Thus:
k = 300 + 330 + 360 + ... + 600.
Here, the interval between successive terms is 30.
Thus:
Number of terms = (600-300)/30 + 1 = 11.
Average = (600+300)/2 = 450.
Sum = 11*450.
Prime-factoring k = (11)(450), we get:
(11)(2*3*3*5*5).
Thus, the greatest prime factor of k is 11.
The correct answer is C.
Last edited by GMATGuruNY on Fri May 30, 2014 8:09 am, edited 1 time in total.
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NOTE: I doubt that the GMAT would use the term "even multiples."prernamalhotra wrote:If integer k is equal to the sum of all even multiples of 15 between 295 and 615, what is the greatest prime factor of k?
1)5
2)7
3)11
4)13
5)17
Yes, this term MAY BE intuitively apparent, but I believe the GMAT test-makers would provide additional text to avoid any ambiguity. Presumably even multiples of 15 are: 30, 60, 90, etc.
In other words, we're looking for multiples of 30
So, k = 300 + 330 + 360 + ... + 570 + 600
Let's examine some terms in this series. . . .
300 = 30(10)
330 = 30(11)
360 = 30(12)
390 = 30(13)
.
.
.
570 = 30(19)
600 = 30(20)
So k = 30(10 + 11 + 12 + ... + 19 + 20)
------------------------------------------------------
Now, let's examine this sum: 10 + 11 + 12 + ... + 19 + 20
Since 20 - 10 + 1 = 11, we know there are 11 numbers to add together.
Aside: A nice rule says: the number of integers from x to y inclusive equals y - x + 1
Since these red numbers are equally spaced (consecutive integers), their sum = (# of values)(average of first and last values)
= [11][(10+20)/2]
= [11][15]
= (11)(15)
-------------------------------------------------
So, k = 30(10 + 11 + 12 + ... + 19 + 20)
= 30(11)(15)
= (2)(3)(5)(11)(3)(5)
We can see that 11 is the greatest prime factor of k
Answer: C
Cheers,
Brent
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k = 300, 330, ...... 600
11/2 ( 2*300 + 10*30)
= 11/2 * (600 + 300) = 450*11
450 = 5x5x2x3x3
So, 11
11/2 ( 2*300 + 10*30)
= 11/2 * (600 + 300) = 450*11
450 = 5x5x2x3x3
So, 11
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