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 primeFor 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:
Cheers,
Brent
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