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# A positive integer is called "square-free" if it has no

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Senior Manager
Joined: 22 May 2003
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A positive integer is called "square-free" if it has no [#permalink]  01 Sep 2003, 16:44
A positive integer is called "square-free" if it has no
factor that is the square of an integer greater than 1.
If n is an even square-free integer, which of the following
must also be square-free?

A)n/2
B)2n
C)n+2
D)n^2
E)None of the above
Manager
Joined: 10 Jun 2003
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A.

If n has no squares as factors, then neither can n/2...because if n/2 did, then n would in the first place...

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Sept 3rd

Senior Manager
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First let us see what a "square free"(SF) integer means. By definition it means that the SF ineteger doesn't has a factor which is sqaure of any integer greater than 1. What that means is that when SF integer written in terms of prime factors than none of the prime factors have power greater than 1.
Example of SF: 2*3*5=30, 5*7=35, 3*7*2=42
Example of Not SF: 2^2*3=12, 2*3^3*5=90

* Now its evident that if any integer n is even SF (must contain 2 as one of its prime factors), n/2 will also be.
some other stuff to ponder over:
* If an interger is even SF, "2n" will NOT be SF. (2^2 factor will come)
* If an integer >1 is odd SF integer, n/2 will not be an Integer.
- Vicks
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Great explanation !! Thanks Vicky

SVP
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I have a slightly different explaination

if N is a even square-free integer then N/2 has to be odd other wise
it would have a factor which is 4 or square of 2
So we can represent N as 2*m * ( 2n+ 1 )

m is odd and n is either odd or even
so N/2 = m * (2n+1 )
m != 2n+1 because N would then have a square of some odd number
So N/2 is also a square-free integer
Senior Manager
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Yet another approach..

Based on the given definition, a "square-free" number does not have 4, 9, 16, 25, ... as one of the factor.

Now n is even "square-free" number
=> so possible values of n = 2, 6, 10, 14,...

Now go through each of the choices.

n/2 = 1, 3, 5, 7, ..... (ALL "square-free")

remaining all choices will have one of the factors from (4, 9, 16, 25, ...)

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