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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
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    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 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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    Thanked by: lheiannie07
    Post 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.

    Thanked by: lheiannie07
    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

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    Thanked by: lheiannie07
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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.

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    Thanked by: lheiannie07
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