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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 Reply

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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
[#permalink]
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

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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

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.

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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!

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

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
[#permalink]
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
[#permalink]
24 Aug 2023, 17:03

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