MHB What should I say about elementary number theory?

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The discussion centers on preparing an engaging talk about elementary number theory, focusing on positive integers and primes, along with their applications in cryptography. A suggested hook for the introduction is the Chinese Remainder Theorem, which is highlighted for its appeal. Additionally, the concept of counting without direct enumeration is presented, with Burnside's Lemma mentioned as a relevant example. This lemma illustrates how to determine the number of distinct arrangements, such as colorings of beads, without tedious counting. Overall, the conversation emphasizes the importance of captivating introductory concepts in presenting elementary number theory.
matqkks
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I need to give an option talk about elementary number theory module. I will discuss how it is study of positive integers particularly the primes and give some cryptography applications. What is a good hook to stipulate in this talk regarding an introduction to elementary number theory?
 
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Chinese Remainder Theorem - always fun. Counting things without actually counting them!
 
tkhunny said:
Chinese Remainder Theorem - always fun. Counting things without actually counting them!
I really like this. Are there any others?
 
matqkks said:
I really like this. Are there any others?

Counting things without actually counting them?
That brings Burnside's Lemma to mind.
It counts for instance the number of different colorings of a string of colored beads - without actually counting them.
 

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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