Positive integers A, B, C are such that C is a multiple of B & B is a

Question

Positive integers A, B, C are such that C is a multiple of B & B is a multiple of A. Is A multiple of 5?

(1) B x C is a multiple of 5
(2) A x B is a multiple of 5

fskilnik · Answer

A,B,C\,\,\, \geqslant 1\,\,\,{	ext{ints}}

\frac{C}{B} = \operatorname{int} \,\,\,\,\,\,\,\,;\,\,\,\,\,\,\,\,\frac{B}{A} = \operatorname{int}

\frac{A}{5}\,\,\,\mathop = \limits^? \,\,\,\operatorname{int}

\left( {1 + 2} ight)\,\,\,\left\{ \matrix{
 {{BC} \over 5} = {\mathop{m int}} \hfill \cr 
 {{AB} \over 5} = {\mathop{m int}} \hfill \cr} ight.\,\,\,\,\,\,\,

\left\{ \begin{gathered}
 \,{	ext{Take}}\,\,\left( {A,B,C} ight) = \left( {1,5,5} ight)\,\,\,\, \Rightarrow \,\,\,\left\langle {{	ext{NO}}} ightangle \,\, \hfill \
 \,{	ext{Take}}\,\,\left( {A,B,C} ight) = \left( {5,5,5} ight)\,\,\,\, \Rightarrow \,\,\,\left\langle {{	ext{YES}}} ightangle \,\, \hfill \ 
\end{gathered} ight.

The correct answer is therefore (E).

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

Regards, 
Fabio.

GMATWhizTeam · Answer

Solution 
 Step 1: Analyse Question Stem 
 • B = n*A, where n is an integer
• C = m*B= m*n*A, where m and n are integers. 
We need to find whether A is a multiple of 5 or not.
 Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE 
 Statement 1:  B*C is a multiple of 5
 •  B*C= n*A*m*n*A = m*n^2*A^2 
• Now,  m*n^2*A^2  is a multiple of 5,
 o A may or may not be a multiple of 5
o Case 1: m or n is a multiple of 5 whereas A is not
   m*n^2*A^2  is a multiple of 5.
 A is not a multiple of 5.  
 o Case 2: A is a multiple of 5 irrespective of m and n
   m*n^2*A^2  is a multiple of 5   
Hence, statement 1 is not sufficient, we can eliminate answer option A and D
 Statement 2:  A*B is a multiple of 5
 •  A*B = n*A^2 
•  n*A^2  is a multiple of 5
 o Case 1: If n is multiple of 5, and A is not
   n*A^2  is a multiple of 5 
o Case 2: If A is a multiple of 5 irrespective of n
   n*A^2  is a multiple of 5 
o A may or may not be a multiple of 5  
Hence, statement 2 is also not sufficient, we can eliminate answer options B 
 Step 3: Analyse Statements by combining. 
From statement 1:  m*n^2*A^2  is a multiple of 5
From statement 2:  n*A^2  is a multiple of 5
By combining the two, we can infer
 • At least one out of n and A is a multiple of 5
 o However, we cannot say for sure whether A is a multiple of 5 or not.   
Hence, the correct answer is  Option E

Positive integers A, B, C are such that C is a multiple of B & B is a

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