Is positive integer n divisible by 4?

Question

(1)  n^2  is divisible by 8

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

Bunuel · Accepted Answer

(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.

Answer: D.

scheol79 · Answer

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.

vitamingmat · Answer

rather randomly picking numbers, follow Bunuel method... Cool stuff

hemanthp · Answer

thanks for the good question and a good explanation!

shrive555 · Answer

The Explanation i had was :

if n is divisible by 4 then n will have atleast two prime factors
n=  2x2 x?x?x?

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

2= n = Sqr/n x Sqr/n
 ( 2 x?x?x? ..) x (  2 x?x?....)
 ( 2x2. ........)
 
can you give example on this please

Bunuel · Answer

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.

utin · Answer

Bunuel, what made you think about :
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???

Bunuel · Answer

Square of an integer is a perfect square --> n is an integer --> n^2 is a perfect square.

JS1290 · Answer

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!

Bunuel · Answer

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.

usmanazeem · Answer

In statement no. 2, why haven't we considered the possibility of n being zero?

Bunuel · Answer

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.

Check for more below threads:
 ALL YOU NEED FOR QUANT ! ! ! 
 Ultimate GMAT Quantitative Megathread

Hope it helps.

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Is positive integer n divisible by 4?

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Is positive integer n divisible by 4?

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