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x^2 is divisible by both 40 and 75. If x has exactly three
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12 Aug 2017, 10:10
12 Aug 2017, 10:10
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\(x^2\) is divisible by both 40 and 75. If x has exactly three distinct prime factors, which of the following could be the value of x?
Indicate all values that apply.
❑ 30
❑ 60
❑ 200
❑ 240
❑ 420
Re: x^2 is divisible by both 40 and 75. If x has exactly three
[#permalink]
09 Jul 2018, 14:44
09 Jul 2018, 14:44
9
Carcass wrote:
\(x^2\) is divisible by both 40 and 75. If x has exactly three distinct prime factors, which of the following could be the value of x?
Indicate all values that apply.
❑ 30
❑ 60
❑ 200
❑ 240
❑ 420
Indicate all values that apply.
❑ 30
❑ 60
❑ 200
❑ 240
❑ 420
Show: :: OA
B, D
\(40 = 2*2*2*5\) and \(75 = 3*5*5\)
For \(x^2\) to be divisible by 40 and 75, its prime-factorization must include at least three 2's (since there are three 2's within 40), at least one 3 (since
there is one 3 within 75), and at least two 5's (since there are two 5's within 75):
\(2^3 * 3^1 * 5^2\)
However, since \(x^2\) is a perfect square, its prime-factorization must have an EVEN NUMBER of every prime factor.
Since the prime-factorization of x must include \(2^3\), \(3^1\) and \(5^2\) -- but \(x\) must have an even number of each of these prime factors -- the least possible option for \(x^2\) is as follows:
\(2^4 * 3^2 * 5^2\)
Since the least possible option for \(x^2 = 2^4 * 3^2 * 5^2\), the least possible option for \(x = 2^2 * 3 * 5 = 60\).
Implication:
\(x\) must be a MULTIPLE OF 60.
In addtion, since \(x\) must have exactly three distinct prime factors, it cannot be divisible by any prime number other than 2, 3 and 5.
Since 30 and 200 are not divisible by 60, eliminate A and C.
Since 420 is divisible by 7 -- a prime number other than 2, 3 and 5 -- eliminate E.
Show: ::
B, D
General Discussion
Re: x^2 is divisible by both 40 and 75. If x has exactly three
[#permalink]
24 Sep 2017, 06:46
24 Sep 2017, 06:46
1
Probably there exist a faster way to solve this question. By the way, I used this one. In order to be a right answer, the square of X must be divisible for both 40 and 75 and X must have only three distinct prime factors.
Thus, I have checked for which of the numbers these two requirements are satisfied and they are for 60 and 240, thus answers B and D!
Thus, I have checked for which of the numbers these two requirements are satisfied and they are for 60 and 240, thus answers B and D!
