Monthly Archives: October 2020

Euclid’s Lemma Proof

Posted on October 31, 2020 by Sumant Sumant

Euclid Continue reading →

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Primitive Roots of Un

Posted on October 31, 2020 by Sumant Sumant

The primitive roots allow us to find if the group is a cyclic one or not.

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Solving Simultaneous Linear Congruence Equations

Posted on October 27, 2020 by Sumant Sumant

In this example we take a look at simultaneous congruence equations. The idea behind this is called Chinese Remainder theorem. Which says if the congruence mods are relative primes then we can construct a number whose solution solves all the … Continue reading →

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Chinese Remainder Theorem (Basic Idea of Proof)

Posted on October 26, 2020 by Sumant Sumant

Chinese Remainder theorem works for multiple linear congruence equations. Here we are going to construct a number which combines several linear congruences which are pair wise relative primes and gives us a unique linear equation which solves all those linear … Continue reading →

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Chinese remainder theorem (example 1 with 2 numbers)

Posted on October 26, 2020 by Sumant Sumant

Here we show how the information of all the residues to two relatively prime number leads to 10 different ordered pairs and each one is related to a specific number.

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Proof: Pick’s Theorem

Posted on October 24, 2020 by Sumant Sumant

Pick’s theorem allows one to quickly found the area of a non interesecting polygon drawn on a grid of lattice points. All one has to do is to count the interior points and the boundary points.

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Relation between congruence equation and its corresponding linear Diophantine equation

Posted on October 22, 2020 by Sumant Sumant

The equation ax= b mod m has a corresponding linear equation ax-my =b. Here we explore the number of solutions to such equation.

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Comment on Wilsons’s theorem and its proof

Posted on October 22, 2020 by Sumant Sumant

Wilson’s theorem is a property about prime number and it hinges on the idea that every number has an inverse in multiplication mod n (obviously 0 is not included). Taking primes > 2. We see that the number of elements … Continue reading →

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Does Zn forms a group under multiplication modulo n when n is not prime ?

Posted on October 22, 2020 by Sumant Sumant

We know that it doesn’t form a group when n is even (because but the first odd number to test is 9 and its . So if the number is not odd prime then it can be factorized by the … Continue reading →

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Example: Solution of Linear Congruence Equation 9x = 12 mod 15

Posted on October 18, 2020 by Sumant Sumant

Here in this example we do a step by step procedure to see how linear congruence equation is solved. We also give the sage code to check your answer.

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