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If n is the smallest integer

This topic has 3 expert replies and 1 member reply

If n is the smallest integer

Post Wed Oct 04, 2017 1:56 pm
If n is the smallest integer such that 432 times n is the square of an integer, what is the value of n?

(A) 2
(B) 3
(C) 6
(D) 12
(E) 24

How can i start the solution to this problem? What is the correct formula in solving it?

OA B

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Post Fri Oct 06, 2017 8:01 am
lheiannie07 wrote:
If n is the smallest integer such that 432 times n is the square of an integer, what is the value of n?

(A) 2
(B) 3
(C) 6
(D) 12
(E) 24

How can i start the solution to this problem? What is the correct formula in solving it?

OA B
Hi lheiannie07,
Let's take a look at your question.

The question states that n is the smallest integer such that 432 times n is the square of an integer.
It means 432n should be a perfect square for the smallest integer n.
Let's try out all the options by starting from option A i.e. 2.

For n = 2
432n = 432(2) = 864
Let's find out if 864 is a perfect square or not by its prie factorization.
864 = (2x2) x (2 x2) x (3x3) x 3
Therefore, it is not a perfect square.

Let's move on to option B now.
For n = 3
432n = 432(3) = 1296
Let's find out if 864 is a perfect square or not by its prie factorization.
1296 = (2x2) x (2 x2) x (3x3) x (3x3)
1296 = (2x2x3x3)^2
1296 = 36^2
Therefore, it is a perfect square of 36.

Hence, Option B is correct.

I am available if you'd like any followup.

_________________
GMAT Prep From The Economist
We offer 70+ point score improvement money back guarantee.
Our average student improves 98 points.

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Top Reply
Post Fri Oct 06, 2017 6:59 am
lheiannie07 wrote:
If n is the smallest integer such that 432 times n is the square of an integer, what is the value of n?

(A) 2
(B) 3
(C) 6
(D) 12
(E) 24
IMPORTANT CONCEPT: The prime factorization of a perfect square (the square of an integer) will have an even number of each prime

For example: 400 is a perfect square.
400 = 2x2x2x2x5x5. Here, we have four 2's and two 5's
This should make sense, because the even numbers allow us to split the primes into two EQUAL groups to demonstrate that the number is a square.
For example: 400 = 2x2x2x2x5x5 = (2x2x5)(2x2x5) = (2x2x5)²

Likewise, 576 is a perfect square.
576 = 2X2X2X2X2X2X3X3 = (2X2X2X3)(2X2X2X3) = (2X2X2X3)²

------NOW ONTO THE QUESTION!!------------------------

Give: 432n is a perfect square

Let's find the prime factorization of 432
We get: 432 = (2)(2)(2)(2)(3)(3)(3)
So, the prime factorization of 432 has four 2's and three 3's
We already have an EVEN number of 2's. So, if we add one more 3 to the prime factorization, we'll have an EVEN number of 3's

So, if n = 3, then 432n = (2)(2)(2)(2)(3)(3)(3)(3)
Since 432n has an EVEN number of each prime, 432n must be a perfect square.

Answer: B

Cheers,
Brent

_________________
Brent Hanneson – Founder of GMATPrepNow.com
Use our video course along with Beat The GMAT's free 60-Day Study Guide

Check out the online reviews of our course
Come see all of our free resources

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Post Fri Oct 06, 2017 8:01 am
lheiannie07 wrote:
If n is the smallest integer such that 432 times n is the square of an integer, what is the value of n?

(A) 2
(B) 3
(C) 6
(D) 12
(E) 24

How can i start the solution to this problem? What is the correct formula in solving it?

OA B
Hi lheiannie07,
Let's take a look at your question.

The question states that n is the smallest integer such that 432 times n is the square of an integer.
It means 432n should be a perfect square for the smallest integer n.
Let's try out all the options by starting from option A i.e. 2.

For n = 2
432n = 432(2) = 864
Let's find out if 864 is a perfect square or not by its prie factorization.
864 = (2x2) x (2 x2) x (3x3) x 3
Therefore, it is not a perfect square.

Let's move on to option B now.
For n = 3
432n = 432(3) = 1296
Let's find out if 864 is a perfect square or not by its prie factorization.
1296 = (2x2) x (2 x2) x (3x3) x (3x3)
1296 = (2x2x3x3)^2
1296 = 36^2
Therefore, it is a perfect square of 36.

Hence, Option B is correct.

I am available if you'd like any followup.

_________________
GMAT Prep From The Economist
We offer 70+ point score improvement money back guarantee.
Our average student improves 98 points.

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Thanked by: lheiannie07
Free 7-Day Test Prep with Economist GMAT Tutor - Receive free access to the top-rated GMAT prep course including a 1-on-1 strategy session, 2 full-length tests, and 5 ask-a-tutor messages. Get started now.
Post Fri Oct 06, 2017 6:59 am
lheiannie07 wrote:
If n is the smallest integer such that 432 times n is the square of an integer, what is the value of n?

(A) 2
(B) 3
(C) 6
(D) 12
(E) 24
IMPORTANT CONCEPT: The prime factorization of a perfect square (the square of an integer) will have an even number of each prime

For example: 400 is a perfect square.
400 = 2x2x2x2x5x5. Here, we have four 2's and two 5's
This should make sense, because the even numbers allow us to split the primes into two EQUAL groups to demonstrate that the number is a square.
For example: 400 = 2x2x2x2x5x5 = (2x2x5)(2x2x5) = (2x2x5)²

Likewise, 576 is a perfect square.
576 = 2X2X2X2X2X2X3X3 = (2X2X2X3)(2X2X2X3) = (2X2X2X3)²

------NOW ONTO THE QUESTION!!------------------------

Give: 432n is a perfect square

Let's find the prime factorization of 432
We get: 432 = (2)(2)(2)(2)(3)(3)(3)
So, the prime factorization of 432 has four 2's and three 3's
We already have an EVEN number of 2's. So, if we add one more 3 to the prime factorization, we'll have an EVEN number of 3's

So, if n = 3, then 432n = (2)(2)(2)(2)(3)(3)(3)(3)
Since 432n has an EVEN number of each prime, 432n must be a perfect square.

Answer: B

Cheers,
Brent

_________________
Brent Hanneson – Founder of GMATPrepNow.com
Use our video course along with Beat The GMAT's free 60-Day Study Guide

Check out the online reviews of our course
Come see all of our free resources

  • +1 Upvote Post
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Thanked by: lheiannie07
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