Magoosh
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.
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 primeAAPL wrote:Magoosh
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.
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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Brent
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Breaking 432 into primes, we have:AAPL wrote:Magoosh
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
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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