Intereting Posts

Graph theory (Graph Connectivity)
There's no continuous injection from the unit circle to $\mathbb R$
How to show that a prime degree separable field extension containing a nontrivial conjugate of a primitive element is Galois and cyclic
Analogy of ideals with Normal subgroups in groups.
Why is important for a manifold to have countable basis?
Finding the inverse of the arc length function
calculate $\sum_{1\le i\le r}\frac{i+1} { r+1}{2r-i\choose r-i}{s+i-2\choose i}+\frac{1}{r+1}{2r\choose r}$?
Locally non-enumerable dense subsets of R
Let $A, B$ be sets. Show that $\mathcal P(A ∩ B) = \mathcal P(A) ∩ \mathcal P(B)$.
Finding a Pythagorean triple $a^2 + b^2 = c^2$ with $a+b+c=40$
How is the Lagrangian related to the perturbation function?
Consequence of the polarization identity?
Maximizing $\frac{x(1-f(x))}{3-f(x)}$
Unique pair of positive integers $(p,n)$ satisfying $p^3-p=n^7-n^3$ where $p$ is prime
Rotating one 3d-vector to another

**Let $a$, $b$, and $c$ be integers, where a $\ne$ 0. Then
$$
$$
(i) if $a$ | $b$ and $a$ | $c$, then $a$ | ($b+c$)
$$
$$
(ii) if $a$ | $b$ and $a$|$bc$ for all integers $c$;
$$
$$
(iii) if $a$ |$b$ and $b$|$c$, then $a$|$c$.**

**Prove that if $a$|$b$ and $b$|$c$ then $a$|$c$ using a column proof that has steps in the first column
and the reason for the step in the second column.**

My book is really vague. I’m not really sure what to do..

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- Applications of Geometry to Computer Science
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- Show $\{0^m1^n | m \neq n\}$ is not regular
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First I was thinking something like this: $a$|$b$ => $b=as$ and $b$|$c$ => $c=bt$ and $a$|$c$ => $c = au$

$$

$$

$b + c + c = b + 2c$

$$

$$

$=as + 2(bt+au)$

Yea then I just got lost.

- Prove or disprove statements about the greatest common divisor
- Not divisible by $2,3$ or $5$ but divisible by $7$
- Existence of uncountable set of uncountable disjoint subsets of uncountable set
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- How to teach mathematical induction?
- How to solve Diophantine equations of the form $Axy + Bx + Cy + D = N$?
- Whats the difference between Antisymmetric and reflexive? (Set Theory/Discrete math)
- What is the proof that the total number of subsets of a set is $2^n$?
- Let $σ$ be the relation on $R×R$ in which $(a,b) σ (x,y)$ if and only if $a ≤ x$ and $b ≤ y$. Prove that $σ$ is a partial order relation.
- Maximum board position in 2048 game

If you are allowed to use fractions then all inferences are immediate consequences of the fact that $\,\Bbb Z\,$ is closed under addition and multiplication – see below. If you are not allowed to use fractions then you can translate the below proofs to integers by clearing denominators, e.g. for $(i)$ we get $\ b=aj,\, c = ak\,\Rightarrow\, a(j + k) = b + c$

$( i)\quad \dfrac{b}a,\,\dfrac{c}a\in\Bbb Z\ \Rightarrow\ \dfrac{b}{a}+\dfrac{c}a = \dfrac{b+c}{a}\in\Bbb Z$

$(ii)\quad \dfrac{b}a,\ c\in\Bbb Z\ \Rightarrow\ c\, \dfrac{b}a = \dfrac{cb}a\in\Bbb Z$

$(iii)\ \ \ \dfrac{b}a,\,\dfrac{c}b\in\Bbb Z\ \Rightarrow\ \dfrac{b}a\dfrac{c}b = \dfrac{c}a\in\Bbb Z$

Note: $ $ if $\,b=0\,$ is allowed in $\,(iii)\,$ then you need to consider that case too.

See this

$$ a|b \implies b = ma \quad \rm and \quad a|c \implies c=na, \quad m,n \in \mathbb{Z} $$

which gives us

$$ b+c = (m+n)a = p a \implies a|(b+c),\quad p=n+m \in \mathbb{Z}. $$

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- Properties of $||x||_X\leq c||Tx||_Y+||Kx||_Z$ for every $x\in X$
- Why does Cantor's Proof (that R is uncountable) fail for Q?