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Math Expert V
Joined: 02 Sep 2009
Posts: 58449
If n is the smallest integer such that 432 times n is the square of an  [#permalink]

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Difficulty:   15% (low)

Question Stats: 78% (01:21) correct 22% (01:43) wrong based on 250 sessions

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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

Kudos for a correct solution.

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Re: If n is the smallest integer such that 432 times n is the square of an  [#permalink]

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2
Here we go:

432 can written as = 2 * 2 * 2 * 2 * 3 * 3 * 3 --> 2^4 * 3^3 ---(1)

so for 432 * n to be a square of an integer, the integer should have even powers to the prime numbers it composed of.

here 2 already has even power -> So n has to be 3 to make the power of 3 in (1) even

Option B is correct
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Re: If n is the smallest integer such that 432 times n is the square of an  [#permalink]

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Answer is B ....

we have 432*n and the problem wants to convert this term to a square number ...

as we know every square number has EVEN POWER in its prime factorization so first of all we should crash 432 to prime factors and then see how many powers it has less to become a

PERFECT square...

432= 4*108 OR 4*9*12 OR : 2^2 * 3^2 * 3* 2^2 and then : 432= 2^4 * 3^3 (16*9=432) as we see here we need only to ONE 3 in order to convert 432 to a perfect square because 2 has

EVEN number of power and 3 does not have ... so if we consider 3 instead of n , we will have 432*3 = 1296

TO check the answer : 36^ 2 =1296 , so this is a perfect square...
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Re: If n is the smallest integer such that 432 times n is the square of an  [#permalink]

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Answer = B = 3

Looking at 432, we can easily say its divisible by 9 (a perfect square)

432 = 9 * 48 = 9 * 16 * 3

16 is already a perfect square, so n = 3
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Re: If n is the smallest integer such that 432 times n is the square of an  [#permalink]

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1
B
432*n=K^2
Prime factorize 432=3^3*2^4
For a number to be perfect square we need odd number of factors ,so 3 must be the value of n
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Re: If n is the smallest integer such that 432 times n is the square of an  [#permalink]

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Bunuel 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

Kudos for a correct solution.

For a number to be square, it should be of the form n^2.
So let's factorise 432 and see how many square terms are present.
432 = 2 * 2 * 2 * 2 * 3 * 3 * 3
= 2^2 * 2^2 * 3^2 * 3
Hence we need one more 3 to make it a perfect square.
Hence option B.

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Math Expert V
Joined: 02 Sep 2009
Posts: 58449
Re: If n is the smallest integer such that 432 times n is the square of an  [#permalink]

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Bunuel 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

Kudos for a correct solution.

MAGOOSH OFFICIAL SOLUTION:

The prime factorization of a square has to have even powers of all its prime factors. If the original number has a factor, say of 7, then when it’s squared, the square will have a factor of 7^2. Another way to say that is: any positive integer all of whose prime factors have even powers must be a perfect square of some other integer. Look at the prime factorization of 432
432 = (2^4)*(3^3)

The factor of 2 already has an even power —- that’s all set. The factor of 3 currently has an odd power. If n = 3, then 432*n would have an even power of 2 and an even power of 3; therefore, it would be a perfect square. Thus, n = 3 is a choice that makes 432*n a perfect square.

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Posts: 4009
Re: If n is the smallest integer such that 432 times n is the square of an  [#permalink]

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Top Contributor
Bunuel 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.

Cheers,
Brent
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Re: If n is the smallest integer such that 432 times n is the square of an  [#permalink]

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Bunuel 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

Breaking 432 into primes, 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.

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Re: If n is the smallest integer such that 432 times n is the square of an  [#permalink]

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Bunuel 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

Kudos for a correct solution.

Just break the 432 down to prime multiple you'll find the answers. THanks Re: If n is the smallest integer such that 432 times n is the square of an   [#permalink] 03 Oct 2018, 00:26
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