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

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Math Expert
Joined: 02 Sep 2009
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If y is an integer, is y^2 divisible by 4?  [#permalink]

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06 Oct 2015, 05:02
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Difficulty:

35% (medium)

Question Stats:

71% (01:26) correct 29% (01:25) wrong based on 153 sessions

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

(1) y is even.
(2) y^3 is divisible by 4.

Kudos for a correct solution.

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Re: If y is an integer, is y^2 divisible by 4?  [#permalink]

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06 Oct 2015, 05:37
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1
1. Since Y is even , it must have at least one factor of 2 .
$$Y^{2}$$= Y*Y
So Y^2 will have at least 2 powers of 2 i.e 2^2 . From each Y , we can factor out a 2. Hence, it will be divisible by 4=2^2

2. $$Y ^ {3}$$ is divisible by 4 =2^2
Y^3 is divisible by 4 .So, Y^3 will contain 2^2 . Since , Y is an integer so each Y should contain at least a single factor of 2.
Hence, $$Y^{2}$$should contain at least 2^2

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Re: If y is an integer, is y^2 divisible by 4?  [#permalink]

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06 Oct 2015, 06:34
1
Bunuel wrote:
If y is an integer, is y^2 divisible by 4?

(1) y is even.
(2) y^3 is divisible by 4.

Kudos for a correct solution.

Question : Is y^2 divisible by 4?

Statement 1: y is even.
If y is even i.e. a multiple of 2 then y^2 will be a multiple of 2^2 i.e. 4, Hence
SUFFICIENT

Statement 2: y^3 is divisible by 4.
Since y^3 is divisible by 4 then y must be Even due to being an Integer hence y^2 will be a multiple of 2^2 i.e. 4, Hence
SUFFICIENT

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Re: If y is an integer, is y^2 divisible by 4?  [#permalink]

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06 Oct 2015, 10:49
1
Bunuel wrote:
If y is an integer, is y^2 divisible by 4?

(1) y is even.
(2) y^3 is divisible by 4.

Kudos for a correct solution.

St1: y is even - Zero is also even so it could be 0 in this case 2,4 in any of those cases it's divisible by 4 SUFFICIENT
St2: y is an even integer in this case because if it were odd odd*odd*odd is odd and not divisible by 4 - SUFFICIEINT

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Math Expert
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Posts: 52279
If y is an integer, is y^2 divisible by 4?  [#permalink]

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11 Oct 2015, 05:40
1
Bunuel wrote:
If y is an integer, is y^2 divisible by 4?

(1) y is even.
(2) y^3 is divisible by 4.

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION:

Question Type: Yes/No. This question asks: “Is y^2 divisible by 4.”

Given information in the question stem or diagram: y is an integer.

Statement 1: y is even. With a conceptual understanding of factors, you know that if y is an even number then y^2 will have to be a multiple of 4. If you want to quickly test this you can use some small even numbers. For example, if y = 2, then y^2=4; yes, that is divisible by 4. If y = 4, then y^2 = 16; this is also divisible by 4. Even 0, which is an even number, will work. 0^2 = 0; and yes, 0 is divisible by 4. This confirms what we already knew from our conceptual understanding: an even
number when squared is a multiple of 4. The answer is either A or D.

Statement 2: y^3 is divisible by 4. For some, this may not seem sufficient at first glance. y^3 might be divisible by 4 while y^2 is not. For example, y could be the cube root of 4. However, if you leverage properly the information in the question stem (always so important!) then you know that “y is an integer.” So y cannot be the cube root of 4; it must be an integer. Once you have established that y must be an integer then the only way for y^3 to be divisible by 4 is for y to be an even number. In other words, it gives you the same information that you already found sufficient in Statement 1. Thus, the correct answer is D.

Note: This is another good example of “When one statement is easy, the other is hard/counterintuitive.” The first statement is relatively simple so you should be extra careful analyzing the second statement. As is often the case, the key with Statement 2 is to make sure you properly leverage the given information.
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Re: If y is an integer, is y^2 divisible by 4?  [#permalink]

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19 Apr 2018, 00:40
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Re: If y is an integer, is y^2 divisible by 4? &nbs [#permalink] 19 Apr 2018, 00:40
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