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
OA B
Source: Magoosh
If n is the smallest integer such that 432 times n is the
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IMPORTANT CONCEPT: The prime factorization of a perfect square (the square of an integer) will have an even number of each primeBTGmoderatorDC 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
OA B
Source: Magoosh
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
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Breaking 432 into primes, we have:BTGmoderatorDC 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
OA B
Source: Magoosh
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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We can create the equation:BTGmoderatorDC 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
OA B
Source: Magoosh
9x = 729
x = 81, so there are 7 * 81 = 567 liters of milk and 81 * 2 = 162 liters of water. Let's let w = the number of liters of water that will be added to make the ratio 7:3, and we have:
567/(162 + w) = 7/3
3 * 567 = 7(162 + w)
3 * 81 = 162 + w
243 - 162 = w
81 = w
Answer: B
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Breaking 432 into primes, we have:BTGmoderatorDC 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
OA B
Source: Magoosh
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
Scott Woodbury-Stewart
Founder and CEO
[email protected]
See why Target Test Prep is rated 5 out of 5 stars on BEAT the GMAT. Read our reviews