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# Integer y equals the sum of all multiples of 21 between 210 and 441

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Joined: 02 Sep 2009
Posts: 47077
Integer y equals the sum of all multiples of 21 between 210 and 441 [#permalink]

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01 Mar 2017, 07:18
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65% (hard)

Question Stats:

59% (01:55) correct 41% (01:43) wrong based on 130 sessions

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Integer y equals the sum of all multiples of 21 between 210 and 441, inclusive. What is the greatest prime factor of y?

A. 7
B. 11
C. 19
D. 31
E. 37

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Re: Integer y equals the sum of all multiples of 21 between 210 and 441 [#permalink]

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01 Mar 2017, 08:03
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Bunuel wrote:
Integer y equals the sum of all multiples of 21 between 210 and 441, inclusive. What is the greatest prime factor of y?

A. 7
B. 11
C. 19
D. 31
E. 37

there are (21-9=12) values that need to be added to find value of y

the sum, y = (210+441)*(12/2) [(First value+Last value)*(No. of terms/2)] in any arithmetic progression

the sum, y = 21*(10+21)*6 = 21*31*6

Biggest prime factor = 31

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Re: Integer y equals the sum of all multiples of 21 between 210 and 441 [#permalink]

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01 Mar 2017, 10:52
Number of multiples of 21=1+(441-210)/2=12

Sum of terms=12(210+441)/2
=2*3*3*7*31

D.

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Re: Integer y equals the sum of all multiples of 21 between 210 and 441 [#permalink]

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01 Mar 2017, 10:59
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Bunuel wrote:
Integer y equals the sum of all multiples of 21 between 210 and 441, inclusive. What is the greatest prime factor of y?

A. 7
B. 11
C. 19
D. 31
E. 37

Multiples of 21 between 210 and 441 = { 210 , 231 , 252....... 420, 441 }

Or, Sum of all multiples 21 between 210 and 441 = 21 ( 10 + 11 ...........20 , 21 )

Or, Sum of all multiples 21 between 210 and 441 = 21*186

Or, Sum of all multiples 21 between 210 and 441 = 21*186 =>3906

Now, $$3906 = 2^1 * 3^2 * 7^1 * 31^1$$

The greatest Prime number of here is 31....

Hence, The correct answer must be (D) 31
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Re: Integer y equals the sum of all multiples of 21 between 210 and 441 [#permalink]

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02 Mar 2017, 05:29
GMATinsight wrote:
there are (21-9=12) values that need to be added to find value of y

Dear GMATInsight,

Can you clarify the above step please?

Thanks
Target Test Prep Representative
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2679
Re: Integer y equals the sum of all multiples of 21 between 210 and 441 [#permalink]

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02 Mar 2017, 18:21
1
1
Bunuel wrote:
Integer y equals the sum of all multiples of 21 between 210 and 441, inclusive. What is the greatest prime factor of y?

A. 7
B. 11
C. 19
D. 31
E. 37

We can first determine the sum of all multiples of 21 between 210 and 441 inclusive using the following formula:

sum = average x quantity

The quantity is (441 - 210)/21 + 1 = 231/21 + 1 = 11 + 1 = 12.

When we have a set of evenly spaced integers, we can calculate the average by using the following formula:

Average = (smallest multiple in the set + largest multiple in the set)/2. Thus:

Sum = (210 + 441)/2 x 12

Sum = 651/2 x 12 = 651 x 6

We need to determine the largest prime factor of 651 x 6.

Since we know the prime factors of 6, let’s prime factorize 651.

651 = 3 x 217 = 3 x 7 x 31. Thus, 31 is the greatest prime factor of y.

Alternative solution:

We are given that y = 210 + 231 + 252 + … + 441. We can factor 21 from each number. That is:

y = 21(10 + 11 + 12 + … + 21)

We can use the formula sum = average x quantity to calculate 10 + 11 + 12 + … + 21 as follows:

10 + 11 + 12 + … + 21 = [(10 + 21)/2] x 12 = 31/2 x 12 = 31 x 6

Thus y = 21(31 x 6) = 3 x 7 x 31 x 2 x 3.

So the largest prime factor of y is 31.

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Re: Integer y equals the sum of all multiples of 21 between 210 and 441 [#permalink]

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30 Mar 2018, 07:37
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Re: Integer y equals the sum of all multiples of 21 between 210 and 441   [#permalink] 30 Mar 2018, 07:37
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