If a and b are positive integers, what is the value of a+b?
(1) a/b = 5/8.
(2) The greatest common divisor of a and b is 1.
The OA C.
Source: GMAT Prep
If a and b are positive integers, what is the value of a+b?
This topic has expert replies
- fskilnik@GMATH
- GMAT Instructor
- Posts: 1449
- Joined: Sat Oct 09, 2010 2:16 pm
- Thanked: 59 times
- Followed by:33 members
Timer
00:00
Your Answer
A
B
C
D
E
Global Stats
Good (conceptual) problem!swerve wrote:If a and b are positive integers, what is the value of a+b?
(1) a/b = 5/8.
(2) The greatest common divisor of a and b is 1.
\[a,b\,\, \geqslant 1\,\,\,{\text{ints}}\,\,\,\left( * \right)\]
\[? = a + b\]
\[\left( 1 \right)\frac{a}{b} = \frac{5}{8}\,\,\,\left\{ \begin{gathered}
\left( {a,b} \right) = \left( {5,8} \right)\,\,\,\, \Rightarrow \,\,\,\,? = 13 \hfill \\
\left( {a,b} \right) = \left( {10,16} \right)\,\,\,\, \Rightarrow \,\,\,\,? = 26 \hfill \\
\end{gathered} \right.\]
\[\left( 2 \right)\,\,a,b\,\,\,{\text{relatively}}\,\,{\text{prime}}\,\,\,\,\left\{ \begin{gathered}
\left( {a,b} \right) = \left( {1,1} \right)\,\,\,\, \Rightarrow \,\,\,\,? = 2 \hfill \\
\left( {a,b} \right) = \left( {1,2} \right)\,\,\,\, \Rightarrow \,\,\,\,? = 3 \hfill \\
\end{gathered} \right.\]
\[\left( {1 + 2} \right)\,\,\,\]
\[\left( 1 \right)\,\,\,\, \Rightarrow \,\,\,\,k\,\,{\text{technique}}\,\,\,\left\{ \begin{gathered}
a = 5k \hfill \\
b = 8k \hfill \\
\end{gathered} \right.\,\,\,\,\,\left( {k \ne 0\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,k \geqslant 1\,\,\operatorname{int} } \right)\]
\[\left[ {{\text{Reason}}\,\,{\text{for}}\,\,k\,\,\operatorname{int} \,\,:\,\,\,k = 2 \cdot \left( {8k} \right) - 3 \cdot \left( {5k} \right) = 2b - 3a = \operatorname{int} - \operatorname{int} = \operatorname{int} } \right]\]
\[\left( 1 \right) + \left( 2 \right)\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,k = 1\,\,\,\, \Rightarrow \,\,\,\,? = a + b = 5 + 8 = 13\,\,\,\, \Rightarrow \,\,\,{\text{SUF}}.\,\,\,\,\,\,\,\]
\[\left( {**} \right)\,\,\,k \geqslant 2\,\,\, \Rightarrow \,\,\,GCD\left( {a,b} \right) = k \ne 1\,\,\, \Rightarrow \,\,\,\left( 2 \right)\,\,{\text{contradicted}}{\text{,}}\,\,\,{\text{impossible}}\,\,\,\,\]
This solution follows the notations and rationale taught in the GMATH method.
Regards,
fskilnik.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
GMAT/MBA Expert
- Jay@ManhattanReview
- GMAT Instructor
- Posts: 3008
- Joined: Mon Aug 22, 2016 6:19 am
- Location: Grand Central / New York
- Thanked: 470 times
- Followed by:34 members
Timer
00:00
Your Answer
A
B
C
D
E
Global Stats
We have a and b are positive integers.swerve wrote:If a and b are positive integers, what is the value of a + b?
(1) a/b = 5/8.
(2) The greatest common divisor of a and b is 1.
The OA C.
Source: GMAT Prep
We have to get the value of a + b.
Let's take each statement one by one.
(1) a/b = 5/8.
Say a = 5k, thus, b = 8k, where k is any integer
=> a + b = 5k + 8k = 13k
Since the value of k is not known, the unique value of a + b cannot be determined. Insufficient.
(2) The greatest common divisor of a and b is 1.
a and b can have many possibilities such as a = 1, b = 2 and a = 2 and b = 3. This will not give the unique value of a + b. Insufficient.
(1) and (2) together
From (1), we have a = 5k and b = 8k and from (2), we have the greatest common divisor of a and b is 1. Since 5 and 8 are co-prime, k must be 1.
So, a = 5k = 5*1 = 5 and b = 8k = 8*1 = 8 => a + b = 5 + 8 = 13. Sufficient.
The correct answer: C
Hope this helps!
-Jay
_________________
Manhattan Review GRE Prep
Locations: GMAT Classes Boston | GRE Prep Course NYC | GRE Prep Dallas | SAT Prep Classes Houston | and many more...
Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.