Tagged Questions
5
votes
3answers
110 views
Prove that $(a+1)(a+2)…(a+b)$ is divisible by $b!$ [duplicate]
The problem is following, prove that:
$$(a+1)(a+2)...(a+b)\text{ is divisible by } b!\text{ for every positive integer a,b}$$
I've tried solving this problem using mathematical induction, but I ...
2
votes
2answers
43 views
Proving that if $a,b$ are even, then $\gcd(a,b) = 2 \gcd(a/2, b/2)$ [duplicate]
Prove that if $a, b$ are both even then $\gcd(a,b) = 2\cdot\gcd(a/2,b/2)$.
Little confused here. I have tried the following but it's basically just repeating the proof unfortunately:
$a = 2 ...
8
votes
3answers
188 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
1answer
53 views
how to prove if $a|b$ and $b\neq 0$, then $|a|\leq|b|$
where the conditions are:
$a \neq 0$, $b \neq 0$ and $a$ and $b$ are integers.
maybe i'm missing something very basic about the properties of an absolute values.
My approach was to supposed, on the ...
3
votes
2answers
980 views
Proof of $\gcd(a,b)=ax+by$
Here is my proof of $\gcd(a,b)=ax+by$ for $a, b, x, y \in \mathbb{Z}$. Am I doing something wrong? Are there easier proofs?
$a,b \in \mathbb{Z}, g=\gcd(a,b)$ and suppose $g \neq ax + by$. Let $c$ be ...
1
vote
3answers
183 views
How to prove that $z\cdot\text{gcd}(a,b)=\text{gcd}(za,zb)$
I need to prove that $z \cdot \text{gcd}(a,b)=\text{gcd}(za,zb)$. I tried a lot, for example, looking at set of common divisors of the two sides, but I can't conclude anything from that. Can you ...
