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Is a^5 divisible by 4?

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Is a^5 divisible by 4?

by Gmat_mission » Mon Apr 23, 2018 12:34 am

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Is a^5 divisible by 4?

(1) a^5 - 8 is divisible by 4.
(2) a is divisible by 6.

[spoiler]OA=D[/spoiler].

How can I show that the second statement is sufficient? May someone helps me? Please.
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Source: — Data Sufficiency |

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by Jay@ManhattanReview » Tue Apr 24, 2018 12:03 am

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Gmat_mission wrote:Is a^5 divisible by 4?

(1) a^5 - 8 is divisible by 4.
(2) a is divisible by 6.

[spoiler]OA=D[/spoiler].

How can I show that the second statement is sufficient? May someone helps me? Please.
We have to find out whether a^5 divisible by 4.

Let's see each statement one by one.

(1) a^5 - 8 is divisible by 4.

=> (a^5)/4 - 8/4 = Integer

=> (a^5)/4 is an integer

=> a^5 divisible by 4. Sufficient.

(2) a is divisible by 6.

=> a is divisible by 2.

=> a^5 divisible by 2^5.

=> a^5 divisible by 4. Sufficient.

The correct answer: D

Hope this helps!

-Jay
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by Brent@GMATPrepNow » Tue Apr 24, 2018 9:34 am

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Gmat_mission wrote:Is a^5 divisible by 4?

(1) a^5 - 8 is divisible by 4.
(2) a is divisible by 6.
Target question: Is a^5 divisible by 4?

Statement 1: a^5 - 8 is divisible by 4.
In other words, a^5 - 8 is a MULTIPLE OF 4
So, we can say that: a^5 - 8 = 4k for some integer k
Add 8 to both sides to get: a^5 = 4k + 8 (for some integer k)
Factor the right side: a^5 = 4(k + 2) (for some integer k)
This tells us that a^5 is a MULTIPLE of 4
Another way to express this is to say that a^5 IS divisible by 4
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: a is divisible by 6
In other words, a is a MULTIPLE OF 6
So, we can say that: a = 6j for some integer j
Rewrite this as: a = (2)(3j)
So, a^5 = (2)(3j)(2)(3j)(2)(3j)(2)(3j)(2)(3j)
Simplify to get: a^5 = (4)(3j)(2)(3j)(2)(3j)(2)(3j)(3j)
This tells us that a^5 is a MULTIPLE of 4
Another way to express this is to say that a^5 IS divisible by 4
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: D

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by Scott@TargetTestPrep » Wed Apr 25, 2018 3:47 pm

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Gmat_mission wrote:Is a^5 divisible by 4?

(1) a^5 - 8 is divisible by 4.
(2) a is divisible by 6.
Statement One Alone:

a^5 - 8 is divisible by 4.

We can rewrite the statement as:

(a^5 - 8)/4 = integer

a^5/4 - 8/4 = integer

a^5/4 - 2 = integer

Since 2 is an integer, a^5/4 must be an integer in order for a^5/4 - 2 = integer. Thus, a^5 must be evenly divisible by 4.

Statement one alone is sufficient to answer the question.

Statement Two Alone:

a is divisible by 6.

We may recall that in order for a number to be divisible by 4 it must contain at least two factors of 2 in its prime factorization.

Since a is divisible by 6 and 6 = 2 x 3, a must have at least one factor of 2 and hence a^5 must have at least five factors of 2. So a^5 is divisible by 4.

Statement two alone is sufficient to answer the question.

Answer: D

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