Question

Make the correct alternative in the following question:

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

(a) 19 (b) 17 (c) 23 (d) 25

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{P}\left(n\right)=3\times 5^{2n+1}+2^{3n+1},\mathrm{for}\mathrm{all}n\in \mathbf{N}. \\ \mathrm{For}n=1, \\ \mathrm{P}\left(1\right)=3\times 5^{2+1}+2^{3+1} \\ =3\times 5^3+2^4 \\ =375+16 \\ =391 \\ =17\times 23 \\ \mathrm{For}n=2, \\ \mathrm{P}\left(2\right)=3\times 5^{4+1}+2^{6+1} \\ =3\times 5^5+2^7 \\ =9375+128 \\ =9503 \\ =17\times 13\times 43 \\ \mathrm{As},\mathrm{HCF}\left(391,9503\right)=17 \\ \mathrm{So},\mathrm{P}\left(n\right)\mathrm{is}\mathrm{divisible}\mathrm{by}17. \end{gathered}$$

Hence, the correct alternative is option (b).