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proof Archives | forthright48 https://forthright48.com/tag/proof/ learning never ends Tue, 05 Nov 2019 15:42:50 +0000 en-US hourly 1 https://wordpress.org/?v=6.5.8 Sum of Co-prime Numbers of an Integer https://forthright48.com/sum-of-coprime-numbers-of-integer/?utm_source=rss&utm_medium=rss&utm_campaign=sum-of-coprime-numbers-of-integer https://forthright48.com/sum-of-coprime-numbers-of-integer/#comments Fri, 07 Dec 2018 19:54:01 +0000 https://forthright48.com/?p=596 Problem Given a number $N$, find the sum of all numbers less than or equal to $N$ that are co-prime with $N$. In case you forgot, a number $x$ is co-prime with $N$ if $gcd(x,N) = 1$. For example, if $N = 10$, then the following numbers are co-prime with it: $[1, 3, 7, 9]$. …

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]]> https://forthright48.com/sum-of-coprime-numbers-of-integer/feed/ 7 GCD Sum Function https://forthright48.com/gcd-sum-function/?utm_source=rss&utm_medium=rss&utm_campaign=gcd-sum-function https://forthright48.com/gcd-sum-function/#respond Sun, 25 Nov 2018 17:38:44 +0000 https://forthright48.com/?p=502 Problem Given a positive integer $N$, find the value of $g(N)$, where $$g(n) = gcd(1,n) + gcd(2,n) + gcd(3,n) + \dots + gcd(n,n) = \sum_{i=1}^n gcd(i,n)$$ For example, $$ \begin{align} g(6) & = gcd(1,6) + gcd(2,6) + g(3,6) + gcd(4,6) + gcd(5,6) + gcd(6,6) \\ & = 1 + 2 + 3 + 2 + …

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]]> https://forthright48.com/gcd-sum-function/feed/ 0 Multiplicative Function https://forthright48.com/multiplicative-function/?utm_source=rss&utm_medium=rss&utm_campaign=multiplicative-function https://forthright48.com/multiplicative-function/#respond Sat, 24 Nov 2018 09:56:53 +0000 https://forthright48.com/?p=439 A function $f(n)$ is called multiplicative function if: $f(n)$ is defined for positive integer $n$. $f(1) = 1$ $f(mn) = f(m)f(n)$ whenever $gcd(m,n) = 1$. For example, the following functions are multiplicative: Euler Phi – $\phi(n)$ Number of Divisor – $\sigma_0(n)$ Sum of Divisor – $\sigma_1(n)$ Divisor Sum Theorem of Multiplicative Function Theorem: Let $f(n)$ …

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]]> https://forthright48.com/multiplicative-function/feed/ 0 Lucas Theorem – Proof and Applications https://forthright48.com/lucas-theorem-proof-and-applications/?utm_source=rss&utm_medium=rss&utm_campaign=lucas-theorem-proof-and-applications https://forthright48.com/lucas-theorem-proof-and-applications/#comments Fri, 16 Nov 2018 08:48:24 +0000 https://forthright48.com/?p=352 Problem Given non-negative integers $N$, $K$ and a prime number $P$, find the value of: $$\binom{N}{K} \mod P$$ Sounds like a simple problem right? There are several ways to solve the problem, depending on the constraints of $N$, $K$, and $P$. Direct: If $N$, $K$ are really small, less than $12$, then we can calculate …

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]]> https://forthright48.com/lucas-theorem-proof-and-applications/feed/ 1 Chinese Remainder Theorem Part 2 – Non Coprime Moduli https://forthright48.com/chinese-remainder-theorem-part-2-non-coprime-moduli/?utm_source=rss&utm_medium=rss&utm_campaign=chinese-remainder-theorem-part-2-non-coprime-moduli https://forthright48.com/chinese-remainder-theorem-part-2-non-coprime-moduli/#respond Sat, 18 Nov 2017 17:39:00 +0000 As promised on the last post, today we are going to discuss the “Strong Form” of Chinese Remainder Theorem, i.e, what do we do when the moduli in the congruence equations are not pairwise coprime. The solution is quite similar to the one we have already discussed in the previous post, so hopefully, it will …

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]]> https://forthright48.com/chinese-remainder-theorem-part-2-non-coprime-moduli/feed/ 0 Euler Phi Extension and Divisor Sum Theorem https://forthright48.com/euler-phi-extension-and-divisor-sum/?utm_source=rss&utm_medium=rss&utm_campaign=euler-phi-extension-and-divisor-sum https://forthright48.com/euler-phi-extension-and-divisor-sum/#comments Sat, 26 Sep 2015 08:57:00 +0000 Previously we learned about Euler Phi Function. Today we are going to look at two theorems related to Euler Phi that frequently appears in CPPS. I am not sure whether these theorems have any official names, so I just made them up. These allow easy references so I will be using these names from now …

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]]> https://forthright48.com/euler-phi-extension-and-divisor-sum/feed/ 2 Euler Totient or Phi Function https://forthright48.com/euler-totient-or-phi-function/?utm_source=rss&utm_medium=rss&utm_campaign=euler-totient-or-phi-function https://forthright48.com/euler-totient-or-phi-function/#comments Fri, 04 Sep 2015 11:38:00 +0000 I have been meaning to write a post on Euler Phi for a while now, but I have been struggling with its proof. I heard it required the Chinese Remainder Theorem, so I have been pushing this until I covered CRT. But recently, I found that CRT is not required and it can be proved …

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]]> https://forthright48.com/euler-totient-or-phi-function/feed/ 4 Extended Euclidean Algorithm https://forthright48.com/extended-euclidean-algorithm/?utm_source=rss&utm_medium=rss&utm_campaign=extended-euclidean-algorithm https://forthright48.com/extended-euclidean-algorithm/#comments Sun, 26 Jul 2015 08:18:00 +0000 https://forthright48.com/2015/07/26/extended-euclidean-algorithm/ Extended Euclidean Algorithm is an extension of Euclidean Algorithm which finds two things for integer $a$ and $b$: It finds the value of $GCD(a,b)$. It finds two integers $x$ and $y$ such that, $ax + by = gcd(a,b)$. The expression $ax + by = gcd(a,b)$ is known as Bezout’s identity and the pair $(x,y)$ that …

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]]> https://forthright48.com/extended-euclidean-algorithm/feed/ 4 Euclidean Algorithm – Greatest Common Divisor https://forthright48.com/euclidean-algorithm/?utm_source=rss&utm_medium=rss&utm_campaign=euclidean-algorithm https://forthright48.com/euclidean-algorithm/#comments Wed, 01 Jul 2015 07:11:00 +0000 https://forthright48.com/2015/07/01/euclidean-algorithm-greatest-common-divisor/ Problem Given two number A and B, find the greatest number that divides both A and B. What we are trying to find here is the Greatest Common Divisor(GCD) of A and B. What is GCD? Like the name suggests, it the largest number ( let that be G ), that can divide both A …

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