Proof

[bosman3321]

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]

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]

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]

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]

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

 

[Steven G]

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.