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

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Re: A positive integer is called "square-Free" if it has no fact [#permalink]
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fozzzy wrote:
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. $$\frac{n}{2}$$
B. $$2n$$
C. $$n + 2$$
D. $$n^2$$
E. none of the above

What are examples of square-free numbers?

Per definition a square-free integer has primes in power of 1. For example 5^3 is NOT square-free because it's a multiple of 5^2.

We are told that n=2k, since n itself is square-free, then k also must be square-free --> n/2=k.

Or just pick some number for n. Say n=2, then among A, B, C, and D, only A (n/2=1) will be square free.

Hope it's clear.
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Re: A positive integer is called "square-Free" if it has no fact [#permalink]
These definition questions tend to be tricky. I usually get thrown off by such questions... do you have any questions for practice?
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Re: A positive integer is called "square-Free" if it has no fact [#permalink]
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Re: A positive integer is called "square-Free" if it has no fact [#permalink]
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Will not pop into too much:

We will do it by hit and trial:

Since 'n' is an even-square free integers.

Then check values of 'n'

n=2 ..... Can be square free
n=4 ... Cannot be since 4 is a factor of 4
n=6.... Can be square free since 6=3*2*1
n=8... Cannot be

Try n=2,6 in answer options and hence (A) it is
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Re: A positive integer is called "square-Free" if it has no fact [#permalink]
Another way to solve this is to take n = 6
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A positive integer is called "square-Free" if it has no fact [#permalink]
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If the Number is a "Square-Free" Number and it is ALSO Even ---

this Means that whatever the Number is, it must ONLY Contain 1 Prime Base = 2

For ex: N = 2 * 3 * 5 = 30, would be an EVEN Square Free Number, because NONE of the Prime Bases are Raised to a Power > 1 (thus, we will not have any Factors that are Perfect Squares) and it has 1 Prime Base of 2

If we divide this Even Square Free Number by 2, it will remove the 1 Prime Base of 2, leaving only the other Prime Bases which already can NOT "create" a Number with a Perfect Square as a Factor.

The remaining Number N after dividing by 2 will be an ODD "Square Free" Number.

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Re: A positive integer is called "square-Free" if it has no fact [#permalink]
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Re: A positive integer is called "square-Free" if it has no fact [#permalink]
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