Proof

bosman3321

New member
Joined
Apr 21, 2019
Messages
3
How can i prove that for all positive integers of m and n: m and n are multiples of each other if and only if m=n
 

pka

Elite Member
Joined
Jan 29, 2005
Messages
8,331
How can i prove that for all positive integers of m and n: m and n are multiples of each other if and only if m=n
If \(\displaystyle n \) is a multiple of \(\displaystyle m \) then \(\displaystyle \exists k\in\mathbb{Z}^+ \) such that \(\displaystyle km=n\).
If \(\displaystyle m \) is a multiple of \(\displaystyle n \) then \(\displaystyle \exists j\in\mathbb{Z}^+ \) such that \(\displaystyle jn=m\).
Please finish the proof an post the result.
 

bosman3321

New member
Joined
Apr 21, 2019
Messages
3
If \(\displaystyle n \) is a multiple of \(\displaystyle m \) then \(\displaystyle \exists k\in\mathbb{Z}^+ \) such that \(\displaystyle km=n\).
If \(\displaystyle m \) is a multiple of \(\displaystyle n \) then \(\displaystyle \exists j\in\mathbb{Z}^+ \) such that \(\displaystyle jn=m\).
Please finish the proof an post the result.
I dont know how to.
 

pka

Elite Member
Joined
Jan 29, 2005
Messages
8,331
I dont know how to.
Suppose that \(\displaystyle m\text{ is a multiple of }n~\&~ n\text{ is a multiple of }m\).
So \(\displaystyle \exists K\in\mathbb{Z}^+[K\cdot n=m]~\&~\exists J\in\mathbb{Z}^+[J\cdot m=n] \)
\(\displaystyle \therefore,\; \)
So it must be the case that:
\(\displaystyle \begin{align*}m&=K\cdot n \\&=K\cdot(J\cdot m)\\&=(K\cdot J)\cdot m \\\large1 &= K\cdot J \end{align*}\)
\(\displaystyle \LARGE\text{Now what?}\)
 

bosman3321

New member
Joined
Apr 21, 2019
Messages
3
If \(\displaystyle n \) is a multiple of \(\displaystyle m \) then \(\displaystyle \exists k\in\mathbb{Z}^+ \) such that \(\displaystyle km=n\).
If \(\displaystyle m \) is a multiple of \(\displaystyle n \) then \(\displaystyle \exists j\in\mathbb{Z}^+ \) such that \(\displaystyle jn=m\).
Please finish the proof an post the result.
Thanks
Got it
 

Jomo

Elite Member
Joined
Dec 30, 2014
Messages
3,658
Thanks
Got it
You were helped showing that if m and n are multiples of another then m=n. Did you show that if m=n, then m and n are multiples of one another? It is extremely easy but needs to be shown.
 
Top