Make the correct alternative in the following question:

Question:

Make the correct alternative in the following question:

For all $n \in \mathbf{N}, 3 \times 5^{2 n+1}+2^{3 n+1}$ is divisible by

(a) 19                        

(b) 17                         

(c) 23                        

(d) 25

Solution:

Let $\mathrm{P}(n)=3 \times 5^{2 n+1}+2^{3 n+1}$, for all $n \in \mathbf{N}$.

For $n=1$,

$\mathrm{P}(1)=3 \times 5^{2+1}+2^{3+1}$

$=3 \times 5^{3}+2^{4}$

$=375+16$

$=391$

$=17 \times 23$

For $n=2$,

$\mathrm{P}(2)=3 \times 5^{4+1}+2^{6+1}$

$=3 \times 5^{5}+2^{7}$

$=9375+128$

$=9503$

$=17 \times 13 \times 43$

As, $\operatorname{HCF}(391,9503)=17$

So, $\mathrm{P}(n)$ is divisible by 17 .

Hence, the correct alternative is option (b).

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