# Tag Info

1 vote

### Is it true that this quadruple sum involving $f(x,y,z,p)=x^p+y^p−z^p$, is equal to the sequence $1,0,0,0,...$ only if $p=1$ or $p=2$?

Outline/long comment: \begin{align*} s(n) &= \sum_{z=1}^n \sum_{y=1}^n \sum_{x=1}^n g \bigl( \gcd (f(x,y,z,p),n) \bigr) \\ &= \sum_{z=1}^n \sum_{y=1}^n \sum_{x=1}^n \sum_{d\mid \gcd (f(x,y,z,p)...
• 85.2k

• 149
Accepted

### What am I doing wrong in evaluating $\sum_{i=1}^{2024} \gcd(i,8)\;$?

I spotted two mistakes: The number of $i\in\{1,\dots,2024\}$ that are relatively prime to $8$ is not $882$ but rather $1012$. Indeed, the integers relatively prime to $8$ are precisely the odd ...
• 85.2k
Accepted

### When do we have $\gcd(a+b, sc)=1$, provided $\gcd(a,c)=1$, $\gcd(b,c)=1$?

As far as I understand, there are two ways to interpret the objective in this problem: finding constant integers $r,s$ such that $\gcd(ra+b,sc) = 1$ for all pairwise coprime positive integers $a,b,c$,...
• 734
1 vote

### show that $F[x]/q(x)= F[x]/p_1(x)\bigoplus \cdots \bigoplus F[x]/p_k(x)$

Let $q=p_1\cdots p_k$, and note that since $p_i$ is irreducible for all $i$, we have that $(p_i)+(p_j)=F[x]$. Indeed, this holds for any polynomial ring over a field see here for instance. Since have ...
• 4,029
1 vote

• 50.5k
### The difference between the HCF and LCM of $x$ and $18$ is $120$. Find $x$.
Let $\alpha=gcd(x,18)$ & $\beta=lcm(x,18)$ Then $\beta=\frac{18x}{\alpha}$ (as $18x=\alpha\beta$) According to the problem the difference between $\beta$ and $\alpha$ is 120 i.e. $\beta-\alpha=120$...
I'm only aware of an extended Euclidean algorithm for elements of an Euclidean domain. $A[x]$ is an Euclidean domain iff $A$ is a field. Regardless, even if there is a more general algorithm that ...