MHB Uncovering Patterns in Questions

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The discussion focuses on identifying patterns in the remainders of powers of two when divided by seven, specifically noting that the remainder is influenced by the exponent's division by three. Participants suggest analyzing the remainders of triangular numbers, represented as k(k+1)/2, when divided by three to uncover further patterns. The conversation emphasizes the importance of modular arithmetic in understanding these relationships. Ultimately, the goal is to establish a clear pattern that can be applied to similar mathematical problems. Understanding these patterns can enhance problem-solving skills in number theory.
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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 $3$.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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