Is the value of 27^a/6^b a prime number? (1) 3a – b = 1 (2) b is a no

Question

Is the value of 27^a/6^b a prime number? (Source: Bell Curves)

(1) 3a – b = 1
(2) b is a non-zero integer.

acegmat123 · Answer

Statement 1:

If b=0, then a=1/3 and 27^a/6^b = 3 ---------------Yes

If b=1, Then a=2/3 and 27^a/6^b = 9/6 = 3/2 -----------No

Insufficient

Statement 2:

If b
eq{0} and b is an integer.

Then 27^a/6^b can never be prime number

Sufficient

IMO B

rohit8865 · Answer

Hi duahsolo Could u please specify the source of your questions/check OA(last 5-6 posts)

??? Per (2) if b>0 then odd/even=non integer---->Not Prime if b<0 then odd*even=even (2 is only even prime,which is never possible as b is integer)----->Not Prime suff Ans B thanks

mbaprep2016 · Answer

answer has to be C.
b = -1 and let a= -1/3
answer can be Prime.

gauravk · Answer

Given = 27^a/6^b

which can be written as = 3^3a/2^b*3^b

= 3^3a-b/2^b

Statement 1: 3a-b = 1

Which leaves us with 3/2^b

We don't have any info about b so if b = 0, we will be left with 3 which is prime. If b =1 we will be left with 3/2.

So statement 1 is insufficient.

Statement 2: b is a non-zero integer, which in itself in not sufficient as it can have any value from -ve to +ve and we don't have any info on a.

= 3^3a-b/2^b

if we put a = -1/3 and b = -1 this will get solved to 2 which is a prime and there can be scenarios when it will not be prime. So this statement itself isn't sufficient.

Taking statement 1 and 2 together.

3^3a-b/2^b

From st 1 we know 3a-b = 1 and st 2 b is a non zero integer. So we know its not a prime.

So option C IMHO

rohit8865 · Answer

Hi mbaprep2016 as per highlighted part if we put b=-1 in statement 1 then a=0 not 1/3

Thanks

duahsolo · Answer

Hi rohit8865,

The source of the question has been added. Thanks for reminding me.

Cheers!

fskilnik · Answer

Beautiful problem duahsolo. (Kudos!)
 \frac{{{{27}^a}}}{{{6^b}}} = \frac{{{3^{3a - b}}}}{{{2^b}}}\,\,\mathop = \limits^? \,\,{	ext{prime}}\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ \begin{gathered}
 \,b = 0\,\,\,{	ext{and}}\,\,\,3a - b = 1 \hfill \
 \,{	ext{OR}} \hfill \
 b = - 1\,\,{	ext{and}}\,\,3a - b = 0 \hfill \ 
\end{gathered} ight.\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ \begin{gathered}
 \,\left( {a\,;\,b} ight)\,\,\mathop = \limits^? \,\,\,\left( {\frac{1}{3};0} ight) \hfill \
 \,{	ext{OR}} \hfill \
 \,\left( {a\,;\,b} ight)\,\,\mathop = \limits^? \,\,\,\left( { - \frac{1}{3}; - 1} ight) \hfill \ 
\end{gathered} ight.

\left( 1 ight)\,\,\,3a - b = 1\,\,\,\,\,\left\{ \begin{gathered}
 \,{	ext{Take}}\,\,\left( {a\,;\,b} ight)\,\, = \,\,\left( {\frac{1}{3};0} ight)\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{	ext{YES}}} ightangle \,\, \hfill \
 \,{	ext{Take}}\,\,\left( {a\,;\,b} ight)\,\, = \,\,\left( {1;2} ight)\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{	ext{NO}}} ightangle \,\, \hfill \ 
\end{gathered} ight.

\left( 2 ight)\,\,\,b 
e 0\,\,\,\operatorname{int} \,\,\,\,\,\,\left\{ \begin{gathered}
 \,{	ext{Take}}\,\,\left( {a\,;\,b} ight)\,\, = \,\,\left( {1;2} ight)\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{	ext{NO}}} ightangle \,\, \hfill \
 \,{	ext{Take}}\,\,\left( {a\,;\,b} ight)\,\, = \,\,\left( { - \frac{1}{3}; - 1} ight)\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{	ext{YES}}} ightangle \hfill \ 
\end{gathered} ight.\,\,

\left( {1 + 2} ight)\,\,\,\left\{ \begin{gathered}
 \,\left( {a\,;\,b} ight)\,\, = \,\,\left( {\frac{1}{3};0} ight)\,\,\,\,{	ext{contradicts}}\,\,\left( 2 ight)\,\, \hfill \
 \,\,\left( {a\,;\,b} ight)\,\, = \,\,\left( { - \frac{1}{3}; - 1} ight)\,\,\,\,{	ext{contradicts}}\,\,\left( 1 ight)\,\, \hfill \ 
\end{gathered} ight.\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{	ext{NO}}} ightangle

The correct answer is (C), indeed.

This solution follows the notations and rationale taught in the GMATH method.

Regards, 
Fabio.

Is the value of 27^a/6^b a prime number? (1) 3a – b = 1 (2) b is a no

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Is the value of 27^a/6^b a prime number? (1) 3a – b = 1 (2) b is a no

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

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My Rewards