# Questions tagged [gcd-and-lcm]

The concepts of *greatest common divisor* (which is also known as *Highest Common Factor*) and *least common multiple* are closely related notions in the integers, and also make sense in certain other rings. The tag is intended to encompass all questions related to these notions.

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### $\gcd(p_{n-1}, \ n^5 - n^3 + n^2 - n + 1) = 1$ where $p_n = n$th prime.

How can I prove in general that, for all $n\geq 2$: $$\gcd(p_{n-1}, \ n^5 - n^3 + n^2 - n + 1) = 1$$ Seems to always be true: ...
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### The Greatest Common Divisor of All Numbers of the Form $n^a-n^b$

For fixed nonnegative integers $a$ and $b$ such that $a>b$, let $$g(a,b):=\underset{n\in\mathbb{Z}}{\gcd}\,\left(n^a-n^b\right)\,.$$ Here, $0^0$ is defined to be $1$. (Technically, we can also ...
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### Generalisation of $\gcd\left(\frac{a^n-b^n}{a-b},a-b\right)=\gcd(n\gcd(a,b)^{n-1},a-b)$ to $\gcd\left(\frac{a^n-b^n}{a-b},a^m-b^m\right)$?

We have the identity $$\gcd\left(\frac{a^n-b^n}{a-b},a-b\right)=\gcd(n\gcd(a,b)^{n-1},a-b).$$ (see here) This appears to be a quite useful result with various applications. I wonder whether there is ...
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### How do I solve this problem involving the LCM of 200 numbers?

Evaluate $x$ if: $$x\cdot\operatorname{lcm}{(102\ldots 200)}=\operatorname{lcm}{(1,2,\ldots 200)}$$ Here's what I have so far, LEMMA 1: In any set of $n$ consecutive positive integers, there must be ...
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### Using Euclidean Algorithm to find GCD of polynomials in $\mathbb{Q}[x]$

I had to find the $\gcd$ of the following polynomials in $\mathbb{Q}[x]$ using the Euclidean algorithm, and I wanted to check that I had done so correctly, since my notes on the subject from class are ...
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### Assume $d\mid n$ and $\gcd(a,d)=1$. Let $t$ be the product of prime numbers that divide $n$, but don’t divide $a$. Show: $\gcd(a+td,n)=1$.

Let $d,n\in\mathbb Z_{>0}$ with $d\mid n$. Let $a\in\mathbb Z$ such that $\gcd(a,d)=1$. Let $t$ be the product of prime numbers that divide $n$, but don’t divide $a$. Show: $\gcd(a+td,n)=1$. ...
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### Distribution and expected value of a random infinite series $\sum_{n \geq 1} \frac{1}{(\text{lcm} (n,r_n))^2}$

Can we find the distribution and/or expected value of $$S=\sum_{n \geq 1} \frac{1}{(\text{lcm} (n,r_n))^2}$$ where $r_n$ is a uniformly distributed random integer, $r_n \in [1,n]$ and $\text{lcm}$...
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### $\sup\left(\frac{\log(\mbox{ lcm }(1,2,\ldots,k))}{k}\right)$ for $k\in \Bbb{Z}, k>1$

In a previous question Asymptotic growth of l.c.m. of all integers below $k$, it was noted that using the Prime Number Theorem you can prove that $$\log(\mbox{ lcm }(1,2,\ldots,k)) =k+\mbox{ o}(k)$$ ...
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### Exploring congruences and identities involving Mersenne primes and the terms of Lucas-Lehmer test

When I was exploring congruences$\dagger$ involving the terms $S_k$ defined in Lucas-Lehmer test (this reference is the Wikipedia, but the main reference is Crandall and Pomerance, Prime Numbers: A ...
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### How can I show that $x \equiv b^{u} \pmod m$ is always a solution to $x^{k} \equiv \pmod m$, even if $\gcd(b,m) \gt 1$?

How can I show that $x \equiv b^{u} \pmod m$ is always a solution to $x^{k} \equiv \pmod m$, even if $\gcd(b,m) \gt 1$? Our method for solving something like $x^k \equiv b\pmod m$ is first to find ...