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# How many different prime factors does N have? (1) 2N has 4

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Director
Joined: 15 Aug 2005
Posts: 796
Location: Singapore
How many different prime factors does N have? (1) 2N has 4 [#permalink]

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05 Oct 2005, 03:15
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How many different prime factors does N have?

(1) 2N has 4 different prime factors.
(2) N ^2 has 4 different prime factors.
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Cheers, Rahul.

Senior Manager
Joined: 30 Oct 2004
Posts: 284

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05 Oct 2005, 06:55
D) ?

1) 2N has four prime factors => N is a product of 3 prime numbers => N has 3 prime factors.
2N can be written as a product of its prime factors => 2^1*a^x*b^y*c^z
where a,b,c are prime.
E.g. 210, 2*3*5*7

2) N^2 has four prime factors => N^2 can be written as a product of its prime factors => a^2*b^2*c^2*d^2 (a,b,c,d are prime)
N has four prime factors.

Don't know if this is correct... and if its right.. how 1) & 2) don't lead to the same answer.
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-Vikram

Current Student
Joined: 28 Dec 2004
Posts: 3357
Location: New York City
Schools: Wharton'11 HBS'12

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05 Oct 2005, 07:29
B for me...

hmm (1) does seem sufficient...

lets see...

say N=3*5*7,
then 2*N will have 4 different prime factors...

3*5*7=105 so 2*105=210 which has roots (7,3, 5, 2) OK good so N has 3 prime factors...

Now suppose N was 210 and 2N is now 420

well 210 has 4 Prime factors, 420 also has 4 DIFFERENT prime factors....

(1) is Insuff

lets see (11)

N^2= well this much simpler, if N^2 has 4 different Prime Factors, N has to have 4 different Prime factors.... Sufficient...

B it is..
Senior Manager
Joined: 30 Oct 2004
Posts: 284

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05 Oct 2005, 07:53
fresinha12 wrote:
B for me...

hmm (1) does seem sufficient...

lets see...

say N=3*5*7,
then 2*N will have 4 different prime factors...

3*5*7=105 so 2*105=210 which has roots (7,3, 5, 2) OK good so N has 3 prime factors...

Now suppose N was 210 and 2N is now 420

well 210 has 4 Prime factors, 420 also has 4 DIFFERENT prime factors....

(1) is Insuff

lets see (11)

N^2= well this much simpler, if N^2 has 4 different Prime Factors, N has to have 4 different Prime factors.... Sufficient...

B it is..

Thanks... I knew I was missing something in 1) ... great explanation as always!
_________________

-Vikram

05 Oct 2005, 07:53
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