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# x^2 is divisible by both 40 and 75. If x has exactly three

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x^2 is divisible by both 40 and 75. If x has exactly three [#permalink]  12 Aug 2017, 10:10
Expert's post
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Question Stats:

33% (01:56) correct 66% (01:41) wrong based on 68 sessions

$$x^2$$ 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
[Reveal] Spoiler: OA

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Re: x^2 is divisible by both 40 and 75. If x has exactly three [#permalink]  24 Sep 2017, 06:46
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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!
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Re: x^2 is divisible by both 40 and 75. If x has exactly three [#permalink]  24 Sep 2017, 14:06
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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

[Reveal] Spoiler: OA
B, D

The factors of 40 = 2*2*2*5 and factors of 75 = 3*5*5

since x^2 is divisible by both 40 and 75

so x must have = $$2^2*3*5$$ = 60. ( numerator should be the LCM of 40 and 75 ie $$2^3*3*5^2$$)

So check the option which is divisible by 60

Only option B and option D satisfy the condition.
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Re: x^2 is divisible by both 40 and 75. If x has exactly three [#permalink]  08 Jul 2018, 06:23
IlCreatore wrote:
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!

I saw that your method by using the calculator of GRE might take more than 1.5 minutes. However, the method in the second post takes less time.
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Re: x^2 is divisible by both 40 and 75. If x has exactly three [#permalink]  08 Jul 2018, 10:33
Expert's post
GRE almost never requires intensive calculation.

In my opinion, you rarely have to use the calc. The fastest way is to rely on your math skills.

Regards
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Re: x^2 is divisible by both 40 and 75. If x has exactly three [#permalink]  09 Jul 2018, 14:44
3
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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

[Reveal] Spoiler: 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.

[Reveal] Spoiler:
B, D

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Re: x^2 is divisible by both 40 and 75. If x has exactly three [#permalink]  07 Jun 2020, 21:32
I thought E is correct 420/60=7
(420^2)/40 = 4410
(420^2)/75 = 2352
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Re: x^2 is divisible by both 40 and 75. If x has exactly three [#permalink]  08 Jun 2020, 04:32
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Expert's post
mageed wrote:
I thought E is correct 420/60=7
(420^2)/40 = 4410
(420^2)/75 = 2352

We're told that x has exactly three distinct prime factors
420 has FOUR distinct prime factors, since 420 = (2)(2)(3)(5)(7)
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Re: x^2 is divisible by both 40 and 75. If x has exactly three   [#permalink] 08 Jun 2020, 04:32
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