Tagged Questions
2
votes
3answers
51 views
Proofs and Number theory
I am needing help proving the following:
For any integer $n$, $n^2$ + 5 is not divisible by $4$
I am aware that an integer $x$ is divisible by integer $y$ if there exists integer $k$ such that ...
2
votes
1answer
52 views
how to show associativity of multiplication for not just 3 operands but for n operands
ie Id like to show
a(bc)=(ab)c
but for any n operands
eg
abcdefg=gfdcabe etc
I can see this is very intuitive that this should be true for all n operands, but as a logical exercise I would like to ...
3
votes
4answers
101 views
Is this a valid proof by induction?
I want to prove by induction that $a^n \mid b^n$ implies that $a \mid b$ holds for all integers $n\geq 1$.
clearly for $n=1$ this is true, since if $a \mid b$, then $a \mid b$.
Suppose this is true ...
1
vote
1answer
100 views
Are there any errors in my proof that only perfect squares have rational square roots?
This is a very simple proof, but I know that proofs which are this simple can often have some erroneous assumptions. Is mine okay?
Argument: With the exception of perfect squares, there are no ...
8
votes
3answers
185 views
Proof of Wolstenholme's theorem.?
According to the theorem :
$$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+...+\frac{1}{p-1} =\frac{r}{q}$$
And we have to prove that $r= 0 \pmod{p^2}$.
(Given $ p>3$, ...
0
votes
4answers
45 views
Prove that if $a$ is a prime, $b_i \in \mathbb{Z}_+$ and $a | \prod_{i = 1}^{n} b_i$ then $a | b_i$ for some $b_i$
A key property of the integers is that: if $\gcd(a,b) = 1$ and $a |bc$, then $a|c$. Use this property to prove that: if $a \in\mathbb{Z}_+$ is prime and $b_i \in \mathbb{Z}_+$ for $1 \leq i \leq n$ ...
2
votes
5answers
111 views
H0w t0 prove that periodic decimal numbers are rational? $a_1…a_k(b_1b_2..b_l)={m \over n}$
Given $a_1...a_k(b_1b_2..b_l)={m \over n}$ how can I prove that periodic decimal numbers are rational?
Where do I even begin?
4
votes
2answers
84 views
How to show that: $\gcd\left( {a^n-b^n \over a-b} ,a-b\right)=\gcd(n d^{n-1},a-b )$
How to show that:
$$ \gcd\bigg( {a^n-b^n \over a-b} ,a-b\bigg )=\gcd(n d^{n-1},a-b ) $$
$a,b\in \mathbb Z$
where $d=\gcd(a,b)$?
$\gcd$ is the greatest common divisor.
0
votes
1answer
121 views
Prove that $p\mid \binom{p}{k},\ 0< k< p$
Prove that:
$$p \,\,\left|\, {p \choose k} \right., \quad 0< k \lt p$$
if $p$ is prime.
how to prove that with direct proof?
0
votes
3answers
113 views
Prove that for any integer $k \ne 0$, $gcd(k, k+1) = 1$
I'm learning to do proofs, and I'm a bit stuck on this one.
The question asks to prove for any positive integer $k \ne 0$, $gcd(k, k+1) = 1$.
First I tried: $gcd(k,k+1) = 1 = kx + (k+1)y$ : But I ...
1
vote
1answer
104 views
A probabilistic method
I am trying to study for a exam and i found a assignmet, that i cant solve.
Consider a board of $n$ x $n$ cells, where $n = 2k, k≥2$. Each of the numbers from $S = \{1,...,\frac{n^2}{2}\}$ is written ...
10
votes
10answers
1k views
Proving $\sqrt 3$ is irrational.
There is a very simple proof by means of divisivility that $\sqrt 2$ is irrational. I have to prove that $\sqrt 3$ is irrational too, as a homework. I have done it as follows, ad absurdum:
Suppose
...
1
vote
3answers
208 views
Advice for a newly minted math major
I just recently decided to go back to school. I had previously majored in piano performance and dropped out to work as a software engineer.
In high school, I did pretty well at math, I got a 5 on ...
0
votes
4answers
3k views
Prove the square root of 7 is irrational using the Division Algorithm and case reasoning
I proved this previously using proof by contradiction like so:
I am not seeing where to start to prove it using the Quotient Remainder theorem or case reasoning however. Can anyone see the best way ...
3
votes
1answer
201 views
Is this a good proof of Wilson's theorem? — ($(n-1)!+1 \equiv_n 0$ iff n is prime)
Theorem: $(n - 1)! + 1 \equiv_n 0$ if and only if $n$ is prime.
To prove that if $n$ is not prime this is not true is trivial, so I'm just interested in proving that this is true for all p:
...
2
votes
1answer
187 views
Proof of max product of partitions of n
For $n \in \mathbb{Z} : n \geq 1$
$
f(n) = \displaystyle\max_{\substack{ x_1+\dotsm+x_k = n\\ x_i\in\mathbb{Z}^{+} }}
x_1 x_2 \dotsm x_k
$
$$
f(n) = \begin{cases}
1 & \text{if ...
