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# If y is the smallest positive integer such that 3,150 multip

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

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Updated on: 17 Dec 2012, 07:53
3
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Difficulty:

25% (medium)

Question Stats:

75% (01:24) correct 25% (01:44) wrong based on 1174 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 GMATD11 on 08 Mar 2011, 05:37.
Last edited by Bunuel on 17 Dec 2012, 07:53, edited 1 time in total.
Renamed the topic and edited the question.
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Re: y - the smallest +ve integer  [#permalink]

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08 Mar 2011, 06:27
6
13
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.

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Re: y - the smallest +ve integer  [#permalink]

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08 Mar 2011, 05:49
1
1
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

You missed 7 as a prime factor of 3150.

3150 = 3,3,5,5,2 and 7
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Re: y - the smallest +ve integer  [#permalink]

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09 Mar 2011, 16:28
2

on prime factorization 3150 can be expressed as 2 *(3^2)*(5^2)*7

to make the above number a perfect square , we know it atleast need to multiply this by 2*7 = 14
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Re: If y is the smallest positive integer such that 3150  [#permalink]

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23 Mar 2012, 20:27
2
1
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 multip  [#permalink]

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17 Jul 2013, 20:09
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 multip  [#permalink]

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17 Jul 2013, 21:26
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 multip  [#permalink]

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10 May 2016, 23:44
3150 = 3 * 1050 = 3*3*350 = 3*3*35*10
= 3*3*5*7*5*2
= (3)^2 * (5)^2 * 14
So we can say
if we want this number to be a perfect square
this has to be multiplied with 14 .
y=14
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Re: If y is the smallest positive integer such that 3,150 multip  [#permalink]

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11 May 2016, 01:02
Logic for a number to be square = all its prime factors should have power 2

3150 = (5^2)(3^2)(2*7)

Only 2 and 7 do not have power 2

to make power 2 multiply with 2*7=14
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Re: If y is the smallest positive integer such that 3,150 multip  [#permalink]

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11 May 2016, 06:24
4
1
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 multip  [#permalink]

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05 Sep 2018, 07:15
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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Re: If y is the smallest positive integer such that 3,150 multip &nbs [#permalink] 05 Sep 2018, 07:15
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