Can someone explain modulo with fractions?

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The discussion explains how to understand the concept of modulo with fractions, specifically focusing on the example of 3^-1 mod 5, which equals 2. It clarifies that in rational numbers, 1/3 is the solution to the equation 3 * x = 1, and similarly, in modular arithmetic, 3 * x = 1 mod 5 also yields x = 2. The relationship between the modular inverse and fractions is highlighted, stating that if gcd(a,m) = 1, then the inverse can be found using the Euclidean algorithm. The conversation encourages further exploration of the topic through resources like the "modular inverse" articles on Wikipedia and Wolfram. Understanding these principles is essential for grasping modular arithmetic with fractions.
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so apparently 3^-1 mod 5 = 2 so (1/3) mod 5 = 2
I don't get how this works, can someone explain?
 
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maxfails said:
so apparently 3^-1 mod 5 = 2 so (1/3) mod 5 = 2
I don't get how this works, can someone explain?

Sure.

In the rational numbers, 1/3 represents the solution to 3 * x = 1. For integers mod 5, we mean the same thing: 3 * x = 1 mod 5. But you can see that 3 * 2 = 1 mod 5, so 3^-1 is just 2.
 
If gcd(a,m) = 1 and ab = 1 (mod m) then b = 1/a (mod m). If you want to find what integer b is congruent to modulo m and you only know a, then you can first use the euclidean algorithm to find it.
 
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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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