turkey-yks 2012 Q10

turkey-yks · Other · lys1-math Number Theory Congruence Reasoning and Parity Arguments
$$\left. \begin{array} { l } 2 ^ { a } \cdot 3 ^ { b } \equiv 0 ( \bmod 12 ) \\ 2 ^ { b } \cdot 3 ^ { a } \equiv 0 ( \bmod 27 ) \end{array} \right\}$$
For positive integers a and b that satisfy both congruences simultaneously, what is the minimum value of the sum $a + b$?
A) 3
B) 4
C) 5
D) 6
E) 7
$$\left. \begin{array} { l } 
2 ^ { a } \cdot 3 ^ { b } \equiv 0 ( \bmod 12 ) \\
2 ^ { b } \cdot 3 ^ { a } \equiv 0 ( \bmod 27 )
\end{array} \right\}$$

For positive integers a and b that satisfy both congruences simultaneously, what is the minimum value of the sum $a + b$?

A) 3\\
B) 4\\
C) 5\\
D) 6\\
E) 7