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

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33 00:00

Difficulty:   15% (low)

Question Stats: 79% (01:21) correct 21% (01:51) wrong based on 1454 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

Originally posted by mrwaxy on 12 Jun 2008, 05:00.
Last edited by Bunuel on 21 Aug 2019, 00:05, edited 4 times in total.
Math Expert V
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Re: If y is the smallest positive integer such that 3,150 multiplied by y  [#permalink]

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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 $$3,150y=(2*3^2*5^2*7)*(2*7)=(2*3*5*7)^2=perfect \ square$$.

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Hope it helps.
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Baten80 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

This question is based on the concept that when you prime factorize a perfect square, the power of every distinct prime factor is even. The reason for this is given in the following blog posts:
http://www.veritasprep.com/blog/2010/12/quarter-wit-quarter-wisdom-writing-factors-of-an-ugly-number/
http://www.veritasprep.com/blog/2010/12/quarter-wit-quarter-wisdom-factors-of-perfect-squares/

3150 is not a perfect square i.e. square of an integer because
$$3150 = 2*3^2*5^2*7$$

Note that 3 and 5 have even powers (2 each) while 2 and 7 have a power of 1 each. Hence to make 3150 a perfect square, we need to make all powers even. So if we multiply 3150 by 2*7, powers of all prime factors will become even.
$$3150*2*7 = 2^2*3^2*5^2*7^2$$ - A perfect square

Similarly, if you had to divide 3150 by the smallest number to get a perfect square, you would have divided it by 2*7 to get rid of 2 and 7 completely so that the remaining prime factors have even powers,
$$3150/(2*7) = 3^2*5^2$$ - A perfect square
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Re: If y is the smallest positive integer such that 3,150 multiplied by y  [#permalink]

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14
GMATD11 wrote:
142.) If y is the smallest positive integer such that 3150 multiplied by y is the square of an integer, then y must be

a) 2
b) 5
c) 6
d) 7
e) 14

upon factoring 3150 i got following prime factors 3,3,5,5 nd 2

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.

Similar questions to practice:

if-m-and-n-are-positive-integer-and-1800m-n3-what-is-108985.html
property-of-integers-104272.html
if-x-and-y-are-positive-integers-and-180x-y-100413.html
number-properties-92562.html
og-quantitative-91750.html
division-factor-88388.html
if-5400mn-k4-where-m-n-and-k-are-positive-integers-109284.html
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Re: If y is the smallest positive integer such that 3,150 multiplied by y  [#permalink]

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2
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 multiplied by y  [#permalink]

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2
2
E.

Create a prime factor tree for 3,150 and see what numbers do not have a pair.
3150 = 5x5x3x3x7x2.... the 7 and 2 do not have a second pair to make it a perfect square of a number. so y must be 7*2 = 14
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Re: If y is the smallest positive integer such that 3,150 multiplied by y  [#permalink]

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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.

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

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Where can I find the logic of how to answer this question?

Is there a section in the math guide that covers this?

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

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1
1
hfbamafan wrote:
Where can I find the logic of how to answer this question?

Is there a section in the math guide that covers this?

Thanks,
Hunter

No special section covers this, but I can recommend similar questions:
if-m-and-n-are-positive-integer-and-1800m-n3-what-is-108985.html
property-of-integers-104272.html
if-x-and-y-are-positive-integers-and-180x-y-100413.html
number-properties-92562.html
og-quantitative-91750.html
division-factor-88388.html
if-5400mn-k4-where-m-n-and-k-are-positive-integers-109284.html

Hope it helps.
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Re: If y is the smallest positive integer such that 3,150 multiplied by y  [#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 multiplied by y  [#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 multiplied by y  [#permalink]

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

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5
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GMATD11 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

Solution:

This problem is testing us on the rule that when we express a perfect square by its unique prime factors, every prime factor's exponent is an even number.

Let’s start by prime factorizing 3,150.

3,150 = 315 x 10 = 5 x 63 x 10 = 5 x 7 x 3 x 3 x 5 x 2

3,150 = 2^1 x 3^2 x 5^2 x 7^1

(Notice that the exponents of both 2 and 7 are not even numbers. This tells us that 3,150 itself is not a perfect square.)

We also are given that 3,150 multiplied by y is the square of an integer. We can write this as:

2^1 x 3^2 x 5^2 x 7^1 x y = square of an integer

According to our rule, we need all unique prime factors' exponents to be even numbers. Thus, we need one more 2 and one more 7. Therefore, y = 7 x 2 = 14

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

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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.

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

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Top Contributor
GMATD11 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.

Cheers,
Brent
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