For all positive integers ' $n$ ' if $3\left(5^{2 \mathrm{n}+1}\right)+2^{3 \mathrm{n}+1}$ is divisible by…
For all positive integers ' $n$ ' if $3\left(5^{2 \mathrm{n}+1}\right)+2^{3 \mathrm{n}+1}$ is divisible by $k$, then the number of prime numbers less than or equal to $k$ is
$17$
$6$
$7$
$8$
Solution
If $n=1$
$3\left(5^{2 \times 1+1}\right)+2^{3 \times 1+1}=3 \times 125+16=391=17 \times 23$
So, $3\left(5^{2 n+1}\right)+2^{3 n+1}$ is divisible by least prime number $k=17$. So, the number of prime numbers less than or equal to 17 is 7 . The numbers are $2,3,5,7,11,13$ and 17 .