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# Is positive integer n divisible by 4?

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Is positive integer n divisible by 4?  [#permalink]

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Updated on: 16 May 2017, 03:45
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61% (01:32) correct 39% (01:36) wrong based on 178 sessions

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Is positive integer n divisible by 4?

(1) $$n^2$$ is divisible by 8

(2) $$\sqrt{n}$$ is an even integer

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Originally posted by shrive555 on 19 Oct 2010, 09:37.
Last edited by Bunuel on 16 May 2017, 03:45, edited 2 times in total.
Renamed the topic and edited the question.
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Re: Is positive integer n divisible by 4?  [#permalink]

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19 Oct 2010, 09:48
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shrive555 wrote:
Is positive integer n is divisible by 4 ?

1) n^2 is divisible by 8
2) sqr/n is even integer.

(1) n^2 is divisible by 8 --> $$n^2=8p=2^3*p$$ --> in order n^2 to be a perfect square p must complete the power of 2 to even number (perfect square has even powers of its prime factors) --> $$n^2=8p=2^3*2q=2^4q$$ --> $$n=\sqrt{2^4q}=4\sqrt{q}$$. Sufficient.

(2) $$\sqrt{n}=2k$$ --> $$n=4k^2$$. Sufficient.

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Re: Is positive integer n divisible by 4?  [#permalink]

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19 Oct 2010, 22:14
D

I solved it quite easily by putting in some numbers to find a pattern, but I love what Bunuel did!!!

Kudos for both Bunuel and the poster.
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Re: Is positive integer n divisible by 4?  [#permalink]

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20 Oct 2010, 00:31
rather randomly picking numbers, follow Bunuel method... Cool stuff
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Re: Is positive integer n divisible by 4?  [#permalink]

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20 Oct 2010, 04:45
thanks for the good question and a good explanation!
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Re: Is positive integer n divisible by 4?  [#permalink]

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20 Oct 2010, 17:29
The Explanation i had was :

if n is divisible by 4 then n will have atleast two prime factors
n= 2x2x?x?x?

1- n^2 = n x n = ( 2x2 x ? x? ) ( 2x2 x? ? )

2= n = Sqr/n x Sqr/n
(2x?x?x? ..) x ( 2x?x?....)
(2x2.........)
Quote:
Bunel : perfect square has even powers of its prime factors

can you give example on this please
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Re: Is positive integer n divisible by 4?  [#permalink]

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21 Oct 2010, 17:36
1
shrive555 wrote:
The Explanation i had was :

if n is divisible by 4 then n will have atleast two prime factors
n= 2x2x?x?x?

1- n^2 = n x n = ( 2x2 x ? x? ) ( 2x2 x? ? )

2= n = Sqr/n x Sqr/n
(2x?x?x? ..) x ( 2x?x?....)
(2x2.........)
Quote:
Bunel : perfect square has even powers of its prime factors

can you give example on this please

$$x$$ is a perfect square means that it's a square of some $$integer$$ $$n$$: $$x=n^2$$, for example 4=2^2, 9=3^2, ... Now, as $$x=n^2$$ then all powers of primes of x must be even (consider $$n=a^p*b^q*c^r$$, where a, b and c are primes of n --> $$x=n^2=a^{2p}*b^{2q}*c^{2r}$$).

Check this for more:
perfect-square-101678.html?hilit=perfect%20square#p799742
a-perfect-square-79108.html?hilit=perfect%20square

Hope it helps.
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Re: Is positive integer n divisible by 4?  [#permalink]

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21 Oct 2010, 18:18
Bunuel wrote:
shrive555 wrote:
Is positive integer n is divisible by 4 ?

1) n^2 is divisible by 8
2) sqr/n is even integer.

(1) n^2 is divisible by 8 --> $$n^2=8p=2^3*p$$ --> in order n^2 to be a perfect square p must complete the power of 2 to even number (perfect square has even powers of its prime factors) --> $$n^2=8p=2^3*2q=2^4q$$ --> $$n=\sqrt{2^4q}=4\sqrt{q}$$. Sufficient.

(2) $$\sqrt{n}=2k$$ --> $$n=4k^2$$. Sufficient.

in order n^2 to be a perfect square p must complete the power of 2 to even number (perfect square has even powers of its prime factors)

why n^2 be a perfect square???
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Re: Is positive integer n divisible by 4?  [#permalink]

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21 Oct 2010, 18:22
utin wrote:
Bunuel wrote:
shrive555 wrote:
Is positive integer n is divisible by 4 ?

1) n^2 is divisible by 8
2) sqr/n is even integer.

(1) n^2 is divisible by 8 --> $$n^2=8p=2^3*p$$ --> in order n^2 to be a perfect square p must complete the power of 2 to even number (perfect square has even powers of its prime factors) --> $$n^2=8p=2^3*2q=2^4q$$ --> $$n=\sqrt{2^4q}=4\sqrt{q}$$. Sufficient.

(2) $$\sqrt{n}=2k$$ --> $$n=4k^2$$. Sufficient.

in order n^2 to be a perfect square p must complete the power of 2 to even number (perfect square has even powers of its prime factors)

why n^2 be a perfect square???

Square of an integer is a perfect square --> n is an integer --> n^2 is a perfect square.
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Re: Is positive integer n divisible by 4?  [#permalink]

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04 May 2017, 17:26
Hi,

Can anyone please help me explain how S2 is sufficient? For example, if you pick 64 as n, you will get 8 and it is divisible by 4. But if you pick 100, you get 10 and it is not divisible by 4. Please help me as I am pretty confused here and not seeing how S2 is sufficient.

Thank You!
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Re: Is positive integer n divisible by 4?  [#permalink]

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05 May 2017, 01:24
csaluja wrote:
Hi,

Can anyone please help me explain how S2 is sufficient? For example, if you pick 64 as n, you will get 8 and it is divisible by 4. But if you pick 100, you get 10 and it is not divisible by 4. Please help me as I am pretty confused here and not seeing how S2 is sufficient.

Thank You!

n in your examples is 64 or 100, not 8 or 10. Both 64 and 100 are divisible by 4. Check complete solution here: https://gmatclub.com/forum/is-positive- ... ml#p802917

Hope it helps.
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Re: Is positive integer n divisible by 4?  [#permalink]

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23 Aug 2018, 23:20
In statement no. 2, why haven't we considered the possibility of n being zero?
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Re: Is positive integer n divisible by 4?  [#permalink]

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23 Aug 2018, 23:27
1
usmanazeem wrote:
In statement no. 2, why haven't we considered the possibility of n being zero?

The question reads: Is positive integer n is divisible by 4 ? 0 is NOT a positive number.

But even for n = 0, the answer to the question whether n is divisible by 4, would be YES, because 0 is divisible by every integer (except 0 itself).

ZERO.

1. 0 is an integer.

2. 0 is an even integer. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without a remainder and as zero is evenly divisible by 2 then it must be even.

3. 0 is neither positive nor negative integer (the only one of this kind).

4. 0 is divisible by EVERY integer except 0 itself.

ALL YOU NEED FOR QUANT ! ! !

Hope it helps.
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Re: Is positive integer n divisible by 4?   [#permalink] 23 Aug 2018, 23:27