Linked Questions

110
votes
7answers
7k views

Why “characteristic zero” and not “infinite characteristic”?

The characteristic of a ring (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. Otherwise, we ...
33
votes
11answers
3k views

Is this $\gcd(0, 0) = 0$ a wrong belief in mathematics or it is true by convention?

I'm sorry to ask this question but it is important for me to know more about number theory. I'm confused how $0$ is not divided by itself and in Wolfram Alpha $\gcd(0, 0) = 0$ . My question here is: ...
21
votes
8answers
26k views

Easiest and most complex proof of $\gcd (a,b) \times \operatorname{lcm} (a,b) =ab.$

I'm looking for an understandable proof of this theorem, and also a complex one involving beautiful math techniques such as analytic number theory, or something else. I hope you can help me on that. ...
16
votes
2answers
6k views

$\gcd(b^x - 1, b^y - 1, b^ z- 1,…) = b^{\gcd(x, y, z,…)} -1$ [duplicate]

Possible Duplicate: Number theory proving question? Dear friends, Since $b$, $x$, $y$, $z$, $\ldots$ are integers greater than 1, how can we prove that $$ \gcd (b ^ x - 1, b ^ y - 1, b ^ z - 1 ,...
17
votes
3answers
1k views

Show $\langle a^m \rangle \cap \langle a^n \rangle = \langle a^{\operatorname{lcm}(m, n)}\rangle$

So I want to show that $\langle a^m \rangle \cap \langle a^n \rangle = \langle a^{\operatorname{lcm}(m, n)}\rangle$. My approach to this problem was to show a double containment, i.e. to show that $\...
1
vote
4answers
2k views

How to use fundamental theorem of arithmetic to conclude that $\gcd(a^k,b^n)=1$ for all $k, n \in$ N whenever $a,b \in$ N with $\gcd(a,b)=1$?

How to use fundamental theorem of arithmetic to conclude that $\gcd(a^k,b^n)=1$ for all $k, n \in$ N whenever $a,b \in$ N with $\gcd(a,b)=1$? Fundamental theorem of arithmetic: Each number $n\geq 2$ ...
6
votes
2answers
2k views

$\mathrm{lcm}(1, 2, 3, \ldots, n)$?

I want to find $\mathrm{lcm}(1, 2, 3, \ldots, n)$ where $2 \le n \le 10^8$ . I'm trying to find a formula . Please Help .
1
vote
4answers
2k views

Show that $abc=[ab,bc,ca]*(a,b,c)=(ab,bc,ca)*[a,b,c]$

Let $a,b \in \mathbb N$. Show that $$ abc=[ab,bc,ca](a,b,c)=(ab,bc,ca)[a,b,c] $$. How would I prove this?
0
votes
2answers
3k views

How I prove the uniqueness of LCM of two non zero integers a, b?

Prove that the least common multiple of two non zero integers is unique. Need to know how to prove the theorem. Thanks.
4
votes
2answers
2k views

Gcd number theory proof: $(a^n-1,a^m-1)= a^{(m,n)}-1$ [duplicate]

Prove that if $a>1$ then $(a^n-1,a^m-1)= a^{(m,n)}-1$ where $(a,b) = \gcd(a,b)$ I've seen one proof using the Euclidean algorithm, but I didn't fully understand it because it wasn't very well ...
4
votes
2answers
317 views

How to show that $\displaystyle [a,b,c] = \frac{abc}{(ab,bc,ca)}$ without prime factorization?

I think this has been asked before, but I couldn't find it on math.SE. I googled it too, but I wasn't lucky enough to find it there either. So, here's the problem: Demonstrate that for any $a,b,c \in ...
2
votes
2answers
807 views

$g^k$ is a primitive element modulo $m$ iff $\gcd (k,\varphi(m))=1$

I'd really love your help with the following one: : Let $g$ a primitive element modulo $m$, $g^{\varphi(m)} \equiv 1\pmod{m}$. I need to prove that $g^k$ is a primitive element modulo $m$ iff $\gcd (...
2
votes
2answers
376 views

Beginner: How to complete the induction case in a proof that all multiples are a product of the least common multiple

As a disclaimer, I'm fairly new to higher-level mathematics and this is my first question here, so please let me know if I need to clarify anything. I am trying to prove that if I have some natural ...
4
votes
3answers
329 views

Constant case CRT: $\,x\equiv a\pmod{\! 2},\ x\equiv a\pmod{\! 5}\iff x\equiv a\pmod{\!10}$

Problem: Find the units digit of $3^{100}$ using Fermat's Little Theorem (FLT). My Attempt: By FLT we have $$3^1\equiv 1\pmod2\Rightarrow 3^4\equiv1\pmod 2$$ and $$3^4\equiv 1\pmod 5.$$ Since $\gcd(2,...
2
votes
1answer
502 views

Ring Theory Least Common Multiples

If $a$ and $b$ are elements in an integral domain with unity 1$\neq$0. Show that $a$ and $b$ have a least common multiple if $a$ and $b$ have a highest common factor. More generally there is a ...

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