Finding patterns in math questions

[SumitKumar]

Please help me to find pattern in this question

 

[Opalg]

Hi SumitKumar, and welcome to MHB!

The remainder when $2^n$ is divided by $7$ depends on the remainder when the exponent $n$ is divided by $3$. So you need to work out the pattern of remainders when numbers of the form $\dfrac{k(k+1)}2$ are divided by $3$.