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Intern  Joined: 25 Jan 2013
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If 5x^2 has two different prime factors, at most how many di  [#permalink]

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Difficulty:   65% (hard)

Question Stats: 46% (01:10) correct 54% (01:10) wrong based on 488 sessions

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If 5x^2 has two different prime factors, at most how many different prime factors does x have?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Originally posted by suprememodelrus on 14 Nov 2013, 09:24.
Last edited by Bunuel on 14 Nov 2013, 09:26, edited 1 time in total.
Renamed the topic and edited the question.
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Re: If 5x^2 has two different prime factors, at most how many di  [#permalink]

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suprememodelrus wrote:
If 5x^2 has two different prime factors, at most how many different prime factors does x have?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

x cannot have more than 2 prime factors, because in this case 5x^2 would also have more than 2 prime factors.

Can x have 2 prime factors? Yes, consider x=2*5=10 (in this case 5x^2 still has 2 prime factors).

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Posts: 38
Re: If 5x^2 has two different prime factors, at most how many di  [#permalink]

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A little bit more explaination:
5x^2 has 2 prime factors.
One of them has to be 5 (since 5 is a factor of 5x^2 and 5 is prime).

Since the question asks for maximum no of prime factors that x can have, the answer has to be 2 (One of the two prime nos has to be 5, like x can be 10 or 15 or 20 ...etc, with 5 along with 2 0r 3 as the other prime factor)
The other factor is not known to us.

Moreover, since the question says 5x^2 has two different prime factors, x must have atleast one factor other than 5
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GRE 1: Q169 V154 Re: If 5x^2 has two different prime factors, at most how many di  [#permalink]

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1
Properties in action->
1->When we multiply a positive integer x with another positive integer n,the number of prime factors of nx may be equal to x or greater than x.
2->$$X$$ and $$X^n$$ always have the exact same prime factors.
Case 1
In this case since 5x^2 has 2 prime factors.
So x can have one prime factors => and that will not be 5 so that 5x^2 has two primes.
OR
Case 2
In this case x can have two prime factors including 5 => e.g => x=5*3 => 5x^2=> 2 prime factors.
Hence At the most -> x can have 2 prime factors.

Hence B.

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Re: If 5x^2 has two different prime factors, at most how many di  [#permalink]

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The tricky moment here is to notice that (5x)^2 has two different prime factors. The author does not mention the real number of prime factors. For example, the number 100 = 2*2*5*5, so has 4 prime factors, but it only has 2 distinct prime factors, 5 and 2.
So in this question, x can have the maximum of 2 prime factors, one can be something like 2,3,7 and the other 5 to maximize the possibilities of having prime factors.
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Re: If 5x^2 has two different prime factors, at most how many di  [#permalink]

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suprememodelrus wrote:
If 5x^2 has two different prime factors, at most how many different prime factors does x have?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Since 5x^2 has two different prime factors, and one of them is 5, then x can have only one prime other than 5. For instance, if x = 10, then 5x^2 = 5*100 = 500; which has two prime factors, namely 2 and 5. So x can have at most two prime factors, i.e., 5 and some other prime

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Re: If 5x^2 has two different prime factors, at most how many di  [#permalink]

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1
X^2 is just a X multiplied by itself.
So it won't increase / decrease the number of prime factors.

for e.g 6 (6=2x3) has two prime factors
36 (36=2x2x3x3) also has two prime factors

Thus, # of prime factors always remains same for x, and x^2

Now coming to the question
5*X^2 has 2 prime factors
Thus 5*X will also have 2 prime factors
So, X can have at most 2 prime factors ( one of which'd be 5)
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Re: If 5x^2 has two different prime factors, at most how many di  [#permalink]

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_________________ Re: If 5x^2 has two different prime factors, at most how many di   [#permalink] 02 Oct 2019, 19:51
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