If f and g are positive integers, is (f + g)/f an integer?

Question

If f and g are positive integers, is  \frac{f + g}{f}  an integer?

(1)  g = 5f − 4

(2)  f = \frac{g}{3}

Kinshook · Answer

Asked: If f and g are positive integers, is  \frac{f + g}{f}  an integer? 
 \frac{f + g}{f} = 1 + \frac{g}{f} 
 \frac{f + g}{f} is an integer if  \frac{g}{f}  is an integer.

(1)  g = 5f − 4 
 \frac{g}{f} = 5 - \frac{4}{f} 
Since it is unknown whether  \frac{4}{f}  is an integer or not
NOT SUFFICIENT

(2)  f = \frac{g}{3} 
 \frac{g}{f} = 3 
is an integer
SUFFICIENT

IMO B

GMATWhizTeam · Answer

Solution 
 Step 1: Analyse Question Stem 
 • f and g are positive integers.
• We need to find if  \frac{f +g}{f} = n , where n is an integer.
 o Or,  f + g = nf ⟹ g = (n-1)f   
Thus, we need to find if g is a multiple of f or not.
 Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE 
 Statement 1:   g=5f−4 
 • From this statement, we can not be sure, if g is a multiple of f or not. For example, consider the below cases:
 o Case 1: if  f = 3 , then  g =5*3 -4 =11 . Here g is not a multiple of f. 
o Case 2: if  f = 1 , then  g =5*1 -4 =1 . Here g is a multiple of f.  
The results of the above two cases are contradictory.
Hence, statement 1 is NOT sufficient and we can eliminate answer Options A and D.
 Statement 2:   f =\frac{g}{3} 
 • According to this statement:  g = 3f 
 o So, g is a multiple of f.   
Hence statement 2 is sufficient.
Thus, the correct answer is  Option B.

If f and g are positive integers, is (f + g)/f an integer?

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