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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

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Question Stats:

\(x^2\) 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
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09 Jul 2018, 14:44

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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

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

Re: x^2 is divisible by both 40 and 75. If x has exactly three
[#permalink]
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!

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

2

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

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.

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!

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.

Re: x^2 is divisible by both 40 and 75. If x has exactly three
[#permalink]
08 Jul 2018, 10:33

Expert Reply

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

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

Regards

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

(420^2)/40 = 4410

(420^2)/75 = 2352

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Joined: **10 Apr 2015 **

Posts: **6218**

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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

1

mageed wrote:

I thought E is correct 420/60=7

(420^2)/40 = 4410

(420^2)/75 = 2352

(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)

Re: x^2 is divisible by both 40 and 75. If x has exactly three
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26 Jul 2021, 11:14

pranab223 420 is divisible by 60 amongst the answer choices too. Can you explain this line: "so x must have = 22∗3∗522∗3∗5 = 60. ( numerator should be the LCM of 40 and 75 ie 23∗3∗5223∗3∗52)." Thank you!

Re: x^2 is divisible by both 40 and 75. If x has exactly three
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19 Sep 2024, 09:07

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Thanks to another GRE Prep Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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19 Sep 2024, 09:07
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