# Tagged Questions

For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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### Let x and y be integers, let x and y be greater than 0. Prove that the gcd (x/gcd(x,y) , y/gcd(x,y) = 1

Very confusing, not really sure how I'm supposed to deduce what $\gcd (x,y)$ is and how $$\gcd \left(\frac{x}{\gcd(x,y)} , \frac{y}{\gcd(x,y)}\right)$$ can be $1$?
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### Suppose $X$ and $Y$ are greater than $0$. Show that $\gcd(X,Y)$ is $1$ iff $\gcd(X^m,Y^m)= 1$

Problem Suppose $X$ and $Y$ are greater than $0$. Show that $\gcd(X,Y)$ is $1$ iff $\gcd(X^m,Y^m)= 1$. Please help with the above. I have no idea what's going on. An explanation would be nice.
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### Fallacious Induction Proof that All Integers are Equal [duplicate]

We're currently learning about induction in my real analysis course. I came up with the following proof that is obviously false but cannot quite figure out why it fails. Here it is... "Any set $S$ of ...
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### Injection function and product of two exponential elements - homomorphisms -

[Fraleigh, p.133, ex. 13.7] Let $f_i: G_i \rightarrow G_1 \times G_2 \times \dots \times G_r$ be given by $f_i(g_i) = (e_1, e_2, ..., g_i, ..., e_r),$ where $g_i \in G_i$ and $e_j$ is the ...
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### Groups Math Proof Help

Show that the indicated set $G$ with the specified operation forms a group by showing that the four axioms in the definition of a group are satisfied. $G = \mathbb Z_5$ under addition mod $5$. I ...
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### Image of Group Homomorphism is Finite and Divides |Domain of Group| - Fraleigh p. 135 13.44

Let $\phi: G \rightarrow G'$ be a homomorphism. Show that if $|G|$ is finite, then $|\phi[G]|$ is finite and divides $|G|$. Because $φ[G] = \{φ(g) \, | \, g ∈ G\}$, we see $|φ[G]| ≤ \quad |G|$ which ...
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### Greatest Common Divisor Proof

If $d = \gcd(a,n)$, must $\dfrac ad$ and $n$ be relatively prime? Prove or disprove. Do I show that they need to be relatively prime and then the inverse that they do not need to be relatively ...
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### Prove that for any two real numbers a and b $\big||a|-|b|\big|< |a-b|$ [duplicate]

I know I should use the triangle inequality.
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### REF(A + B) = REF(A) + REF(B) [Strang P130 3.3.5]

Describe all $m$ by $n$ matrices $A$ and $B$ such that $ref(A) + ref(B) = ref(A + B)$. Is it true that $ref(A) = A$ and $ref(B) = B$? Does $ref(A - B) = rref(A - B)$? Here, ref = Row Echelon ...
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### Counterexamples to Nonidentities - Power of Cosets and Right Coset - Fraleigh p. 103 10.30, 33

Let $H \le$ group G and $a, b \in G.$ Prove or give a counterexample. If $aH= bH,$ (30.) then $Ha= Hb.$ (33.) then $a^2 H = b^2 H.$ I understand p. 3: Let $G = S_3$ and $H = \{(1), (1,3)\}$. ...
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### Fixed point and period of continuous function

Prove/ Disprove: Let $f:(0,1)\to(0,1)$ be such that $|f(x)-f(y)|\leq 0.5|x-y|$ for all $x ,y.$ Then f has a fixed point. 2.Let $f:\mathbb R\to\mathbb R$ be continuous and periodic with period ...
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### Show that $z$ is prime if $z|xy$ implies $z|x$ or $z|y$

Let $z$ be an integer greater than or equal to $2$. Suppose for all integers $x$ and $y$ that $z|xy$ implies $z|x$ or $z|y$. Show that $z$ is prime.
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### Let $x$ be greater than $1$. Prove $x$ is prime if and only if for every integer $y$, either $\gcd(x,y)=1$ or $x\mid y$.

I've been having serious trouble with this problem, The first direction-> Proving x is prime if for every integer y, either gcd(x,y)=1 or x|y doesn't seem too difficult. We know that if gcd(x,y)=1 ...
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### What does “possible to define” mean?

What does "possible to define" mean in general? First I thought it means that "can not lead to a contradiction", but such seems to be hard to prove. Then for the proof I was looking at, involving ...
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### Using an automatic tool for checking geometric conjectures

I do a lot of research about squares, and I thought of using some automatic tool for proving / disproving some geometric conjectures. As a simple example, consider the following Square coloring ...
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### Verification and help to simplify an argument about closure of some sets.

Hi everyone I'd like to know if what I have so far is correct, I think is much work for something which is too simple I would appreciate any advice or whatever. Moreover, I have doubt in (3) and (4), ...
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### Basic proof problem from “How to Prove it A Structured Approach”

I got the book How to Prove it A Structured Approach and I'm ashamed to admit I failed to even do the first problem in the introduction chapter: a) Factor $2^{15} - 1 = 32767$ into a product of two ...
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### Analytic continuation of zeta is meromorphic on $\mathbb{C}$ with simple pole at 1

We have the following identity: For some contour $\gamma$ and $\forall s \in \mathbb{C}$ Re $s > 1$: $$-2i\sin(\pi s) \Gamma(s)\zeta(s)= \Large\int_{\gamma} \frac{(-z)^{s-1}}{e^z-1}dz$$ The ...
I am studying a proof of a theorem and in the proof i have the following equality  \int_{0}^{\pi /2} \frac{\left|\sin(2nt)\right|}{t} \ \mathrm dt= \sum_{k=1}^{n} \int_{0}^{\pi} \frac{\sin(t)}{ t ...