Is a^5 divisible by 4?

Question

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

rohit8865 · Answer

(1) a^5 – 8==(x-2)4.......where a^5=x*4......
thus a^5 is div by 4...........suff

(2) a=6=2*3

a^5=2^5*3^5
thus a^5 is div by 4.......suff

Ans D

BrentGMATPrepNow · Answer

Target question :    Is a⁵ divisible by 4?

Statement 1 : a⁵ – 8 is divisible by 4.  
In other words, a⁵ – 8 is a MULTIPLE OF 4
So, we can say that: a⁵ – 8 = 4k for some integer k
Add 8 to both sides to get: a⁵ = 4k + 8 (for some integer k)
Factor the right side: a⁵ =  4 (k + 2) (for some integer k)
This tells us that a⁵ is a MULTIPLE of  4 
Another way to express this is to say that  a⁵ 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, a⁵ = (2)(3j)(2)(3j)(2)(3j)(2)(3j)(2)(3j)
Simplify to get: a⁵ = ( 4 )(3j)(2)(3j)(2)(3j)(2)(3j)(3j)
This tells us that a⁵ is a MULTIPLE of  4 
Another way to express this is to say that  a⁵ IS divisible by 4 
Since we can answer the  target question  with certainty, statement 2 is SUFFICIENT

Answer: D

Cheers,
Brent

CrackverbalGMAT · Answer

Solution:

St(1):- a^5 – 8 is divisible by 4

=>8 is already divisible by 4 so a^5 is also divisible  (Sufficient)

St(2):- a is divisible by 6

=>a=6k where K is an integer

=> a^5 = 6k^5 = (3x2k)^5

=>2^5 = 32 is divisible by 4

=>a^5 is divisible  (Sufficient)

(option d)
 
  Devmitra Sen
SME GMAT

Kimberly77 · Answer

Hi Brent  BrentGMATPrepNow , here in St 1, a⁵/4 = (k + 2).
In St 2, a⁵/4 = (3j)(2)(3j)(2)(3j)(2)(3j)(3j). So does these not contradict that both statements are not the same answers? Or do we only care to be able to derive a⁵/4 here only whreas (k + 2) or (3j)(2)(3j)(2)(3j)(2)(3j)(3j) doesn't matter here? Thanks Brent

BrentGMATPrepNow · Answer

I'll answer your question with a question: Can 2x = y? 
Yes, there are values of x and y such that 2x = y (even though 2x and y look different).

Similarity, just because (k + 2) and (3j)(2)(3j)(2)(3j)(2)(3j)(3j)  look  different, doesn't mean they are different. 
In fact, since both expressions equal a⁵/4, they must be equal.

Archit3110 · Answer

For a^5 to be divisible by 4 a has to be even 
#1
a^5-8 divisible by 4
a has to be even and a least is 2 sufficient
#2
a is divisible by 6
a is 2*3*x where x is any number 
a^5 will be divisible by 4 as it has 2
Sufficient
Option D

Posted from my mobile device

Kimberly77 · Answer

I see thanks BrentGMATPrepNow

ScottTargetTestPrep · Answer

Statement One Alone:

\Rightarrow  a^5 – 8 is divisible by 4.

It follows that a^5 - 8 = 4k for some integer k. Then, a^5 = 4k + 8 = 4(k + 2). This means that a^5 is even, which in turn means that a is even. If a is even, then a^5 is divisible by 4 (and in fact, a^5 must be divisible by 32). Statement one alone is sufficient.

Eliminate answer choices B, C, and E.

Statement Two Alone:

\Rightarrow  a is divisible by 6.

Thus, a = 6s for some integer s. Then, a^5 = (6s)^5 = 2^5 * (3s)^5 = 32 * (3s)^5 is divisible by 4. Statement two alone is sufficient.

Answer: D

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Is a^5 divisible by 4?

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