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Intern  Joined: 17 May 2008
Posts: 12
If y is the smallest positive integer such that 3,150  [#permalink]

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3
26 00:00

Difficulty:   15% (low)

Question Stats: 78% (01:23) correct 22% (01:48) wrong based on 1496 sessions

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If y is the smallest positive integer such that 3,150 multiplied by y is the square of an integer, then y must be

A. 2
B. 5
C. 6
D. 7
E. 14

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Originally posted by mrwaxy on 12 Jun 2008, 05:00.
Last edited by Bunuel on 12 Jun 2013, 04:33, edited 3 times in total.
Added the OA
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Math Expert V
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Posts: 55266
Re: If y is the smallest positive integer such that 3,150  [#permalink]

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9
12
mrwaxy wrote:
If y is the smallest positive integer such that 3,150 multiplied by y is the square of an integer, then y must be
A. 2
B. 5
C. 6
D. 7
E. 14

Detailed explanation would be appreciated.

$$3,150=2*3^2*5^2*7$$, now $$3,150*y$$ to be a perfect square $$y$$ must complete the odd powers of 2 and 7 to even number (perfect square has even powers of its primes), so the least value of $$y$$ is 2*7=14. In this case $$3,150y=(2*3^2*5^2*7)*(2*7)=(2*3*5*7)^2=perfect \ square$$.

Answer: E.

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Manager  Joined: 29 Aug 2005
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Re: PS - S26 q30  [#permalink]

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IMO E should be the answer for this

i just tried plugging in the numbers and found out that

14*3150 = 44, 100, which is a square of 210

HTH
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Re: If y is the smallest positive integer such that 3,150  [#permalink]

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1
2
answer E. factor out the number and find any prime numbers that are not paired. 7 & 2.
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Re: If y is the smallest positive integer such that 3,150  [#permalink]

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factorise 3150, to find out the missing doubles... 3150 = 5x5x3x3x2x7... so 2x7=14... when multiplied to 3150, will make it a perfect square... answere is E
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Re: If y is the smallest positive integer such that 3,150  [#permalink]

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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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Re: If y is the smallest positive integer such that 3,150  [#permalink]

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1
mrwaxy wrote:
If y is the smallest positive integer such that 3,150 multiplied by y is the square of an integer, then y must be

A. 2
B. 5
C. 6
D. 7
E. 14

In such questions we need to break the number into the smallest possible prime factors. So the smallest prime factors of 3150 are:
315*10=63*5*2*5=7*9*5*2*5=7*3*3*5*2*5. In order to get a square of an integer we have to have at least two identical primes. In our case we have 3*3 and 5*5 corresponding to this condition but not 2*7 so our smallest number should be 14.

Answer is E
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Re: if y is the smallest positive interger such that 3150 multip  [#permalink]

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1
kumar83 wrote:
if y is the smallest positive interger such that 3150 multiplied by y is the square of an interger, that Y must be

A) 2
B) 5
C) 6
D) 7
E) 14

Kindly Explain.

3150 =$$2*3^2*5^2*7$$
For it to be perfect square all the prime number should be least raised to the power 2
in 3150 ...only 2 and 7 needs to be multiplied so that all prime will be raised power 2
hence least value of $$4y = 2*7 = 14$$

hence E
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Re: If y is the smallest positive integer such that 3,150  [#permalink]

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Hello,

can anyone help me with this type of question? I don't get it why the remaining numbers, 7 and 2, are the smallest positive integer y. Which chapter in the MGMAT books should i restudy to deal with this kind of problem? I don't understand the explanation in the OG which says: "To be a perfect square, 3,150y must have an even number of each of its prime factors."

Any help is appreciated!
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Re: If y is the smallest positive integer such that 3,150  [#permalink]

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Re: If y is the smallest positive integer such that 3,150  [#permalink]

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3150 = 5*5*3*3*2*7

To be a perfect square y needs to be 2*7=14
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Re: If y is the smallest positive integer such that 3,150  [#permalink]

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Top Contributor
mrwaxy wrote:
If y is the smallest positive integer such that 3,150 multiplied by y is the square of an integer, then y must be

A. 2
B. 5
C. 6
D. 7
E. 14

Key concept: The prime factorization of a perfect square (the square of an integer) will have an EVEN number of each prime.
For example, 36 = (2)(2)(3)(3)
And 400 = (2)(2)(2)(2)(5)(5)

Likewise, 3150y must have an EVEN number of each prime in its prime factorization.
So, 3150y = (2)(3)(3)(5)(5)(7)y
We have an EVEN number of 3's and 7's, but we have a single 2 and a single 7.
If y = (2)(7), then we get a perfect square.

That is: 3150y = (2)(2)(3)(3)(5)(5)(7)(7)

So, if y = 14, then 3150y is a perfect square.

Answer:

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Re: If y is the smallest positive integer such that 3,150  [#permalink]

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I wanted to share my GMAT Timing Tips for this question (the links below include growing lists of questions that you can use to practice these tips):

Prime factors of a perfect square, perfect cube, etc.: As others in this thread have pointed out, we need to know that all prime factors of a perfect square have exponents that are even. If we find that any of the prime factors of 3,150 do not have even exponents, y will need to contain each of those prime factors, so that the prime factorization of 3,150*y will have an even exponent for each of those prime factors.

Prime factorization: In order to determine the prime factors of y, we need to do the prime factorization of 3,150, so let's try to do it as efficiently as possible. Because factors of 10 and 5 are easy to see, I recommend starting by factoring 3,150 into 315*10, then 63*5*2*5. We can also recognize that 63 = 9*7 = 3^2 *7. This means that the prime factorization of 3,150 is 2 * 3^2 * 5^2 * 7. Since there are odd powers of 2 and 7, y must contain factors of 2 and 7, and the smallest possible value of y is 2*7 = 14.

Please let me know if you have any questions, or if you would like me to post a video solution!
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Please read and share my free GMAT Timing Strategy Guide! Re: If y is the smallest positive integer such that 3,150   [#permalink] 26 Aug 2018, 00:30
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