If \(2 \cdot 4^{2 k+1}+3^{3 k+1}=11 t\) and \(2 \cdot 4^{2 k+3}+3^{3 k+4}=11\) \(\left(p t+3^q\right)\),…
If \(2 \cdot 4^{2 k+1}+3^{3 k+1}=11 t\) and \(2 \cdot 4^{2 k+3}+3^{3 k+4}=11\) \(\left(p t+3^q\right)\), where \(k, t \in Z^{+}\), then \((p, q)=\)
\((16,3 k+1)\)
\((16,3 k+4)\)
\((32,3 k+1)\)
\((32,3 k+4)\)
Solution
By verification method, if \(k=0\)
Then, \(2 \cdot 4^1+3^1=11 t\)
\(\Rightarrow \quad 8+3=11 t \Rightarrow t=1\)
Now, from the second given relation on putting the values of \((k, t)=(0,1)\), we get
\(\begin{aligned}
2 \cdot(64)+81 & =11\left(p+3^q\right) \\
\Rightarrow \quad 128+81 & =11\left(p+3^4\right) \\
\Rightarrow \quad 11\left(p+3^q\right) & =209 \Rightarrow p+3^q=19
\end{aligned}\)
Now, from the option \(p\) must be 16 and \(q=1\)
\(=(3 k+1)_{k=0} .\)
Hence, option (a) is correct.