2025/025) How many numbers (n) are there between 1 and 200 such that $\frac{n}{2}$ , $\frac{n}{3}$ ,$\frac{2n+1}{5}$ are all composite natural numbers (CAT 2020)?

First let us find the number such that $\frac{n}{2}$ ,  $\frac{n}{3}$ ,$\frac{2n+1}{5}$ are integers

as  $\frac{n}{2}$ ,  $\frac{n}{3}$ are integers so $\frac{n}{6}$ is integer say k

so n= 6k

As in denominator we have 2 3 and 5 so we should have mod 30

take k from 0 to 5 we get n = 12 satisfies  $\frac{2n+1}{5}$ integer

so we get values n =12 , 42, 72, 102,132,162,192 .

as n is multiple of 6 $\frac{n}{2}$ ,  $\frac{n}{3}$ are composite we need to check   $\frac{2n+1}{5}$ composite . let us compute $\frac{2n+1}{5}$

n = 12 gives 5 so no

n = 42 gives 17 so no

n = 72 gives 29 no

n =  102 gives 41 no

n = 132 gives 53 no

 n = 162 gives 65 yes

n = 192 gives 77 yes

so there are 2 values