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If n is the smallest integer

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by BTGmoderatorDC » 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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by Jay@ManhattanReview » Wed Oct 04, 2017 9:44 pm
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
So we gave 432n a perfect square, thus, √(432n) is an integer.

Let's factorize 432n.

432n = 2*216n = (2^2)*108n = (2^3)*54n = (2^4)*27n = (2^4)*(3^3)n

Thus, √(432n) = √[(2^4)*(3^3)n] = 2^2*√[(3^3)n]

Since the exponent of 3 is 3, an odd number, we can't have its square root an integer. Thus, to make (3^3)*n a perfect square, n must be one among 3^1, 3^3, 3^5, 3^7, ..., 3^(an odd integer). The minimum value of n = 3^1 = 3.

The correct answer: B

Hope this helps!

-Jay

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by vaibhav101 » Thu Oct 05, 2017 7:29 am
When you are not able to solve a particular problem then, look for options from the answer choices,
we have to find the least value of n for which 432n will be a perfect square.
if n=2, then 432 x 2=864not a perfect square
if n=3, then 432 x 3=1296which is the square of 36
therefore answer is n=3.
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by Brent@GMATPrepNow » 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
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by EconomistGMATTutor » 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.

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Re: If n is the smallest integer

by Scott@TargetTestPrep » Fri Jan 24, 2020 3:19 pm
BTGmoderatorDC wrote: ↑
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
Breaking 432 into its prime factors, we have:

432 = 8 x 54 = 8 x 9 x 6 = 2^4 x 3^3. A perfect square always has prime factors with even exponents, so, in order for 432 x n to be a perfect square, we need one more prime factor of 3. Thus, n is 3.

Answer: B

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