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# If y is an integer, is y^2 divisible by 4?

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Intern
Joined: 18 Jan 2012
Posts: 22
If y is an integer, is y^2 divisible by 4? [#permalink]

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07 Feb 2012, 20:41
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25% (medium)

Question Stats:

75% (00:30) correct 25% (00:42) wrong based on 88 sessions

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If y is an integer, is y^2 divisible by 4?

(1) y is divisible by 4
(2) y is divisible by 6
[Reveal] Spoiler: OA

Last edited by Bunuel on 08 Feb 2012, 02:16, edited 1 time in total.
Edited the question
Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 4678
Re: Number Properties - Question 2 [#permalink]

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07 Feb 2012, 23:09
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Hi, there. I'm happy to help with this.

Prompt: if y is an integer , is y^2 divisible by 4?

Fact #1: In order for an integer N to be divisible by 4, N must have at least two factors of 2 in its prime factorization.

Fact #2: When you square a number, say T^2, whatever prime factors are in the prime factorization of T are doubled in the prime factorization of T^2. Say a particular prime factor appears three times in T --- then it will appear six times in T^2.

Fact #3: An even number, by definition, has at least one factor of 2 in its prime factorization.

Therefore, the square of any even number will have at least two factors of 2, and therefore will be divisible by 4.

The question "is y^2 divisible by 4?" is entirely equivalent to the question "is y an even integer?"

All of that was the mathematical heavy-lifting for the question. Now, the statements will be a piece of cake.

Statement #1: y is divisible by 4. Therefore y is even. Sufficient.

Statement #2: y is divisible by 6. Therefore y is even. Sufficient.

Both statements sufficient. Answer = D

Does all that make sense?

Here's another odd/even DS question for practice.

http://gmat.magoosh.com/questions/868

Please let me know if you have any questions.

Mike
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Mike McGarry
Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)

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Posts: 44412
Re: Number Properties - Question 2 [#permalink]

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08 Feb 2012, 02:16
If y is an integer, is y^2 divisible by 4?

(1) y is divisible by 4 --> as y itself is divisible by 4 then y^2 will also be divisible by 4. Sufficient.
(2) y is divisible by 6 --> y=6k, for some integer k --> y^2=36k^2=4*(9k^2) --> we can see that y^2 has 4 as a factor hence it is divisible by 4. Sufficient.

Hope it's clear.
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If y is an integer, is y^2 divisible by 4? [#permalink]

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26 Nov 2017, 11:28
nimc2012 wrote:
If y is an integer, is y^2 divisible by 4?

(1) y is divisible by 4
(2) y is divisible by 6

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 1 variables and 0 equation, we need to consider each condition one by one.

Condition 1)
$$y = 4n$$ for some integer $$n$$.
$$y^2 = 16n^2$$
Since $$16$$ is divisible by $$4$$, this is sufficient.

Condition 2)
$$y = 4m$$ for some integer $$m$$.
$$y^2 = 36m^2$$
Since $$36$$ is divisible by $$4$$, this is sufficient.

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
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If y is an integer, is y^2 divisible by 4?   [#permalink] 26 Nov 2017, 11:28
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