MHB Uncovering Patterns in Questions

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SUMMARY

The discussion focuses on identifying the pattern of remainders when \(2^n\) is divided by \(7\), specifically noting that this pattern is contingent upon the exponent \(n\) modulo \(3\). It emphasizes the necessity of analyzing the remainders of triangular numbers, expressed as \(\frac{k(k+1)}{2}\), when divided by \(3\). This mathematical exploration reveals a systematic approach to understanding modular arithmetic in relation to powers of \(2\).

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Please help me to find pattern in this question
Screenshot_2020-09-09-13-04-07-109.jpeg
 
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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 $3$.
 

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