Proof

bosman3321

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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
 
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 n\displaystyle n is a multiple of m\displaystyle m then kZ+\displaystyle \exists k\in\mathbb{Z}^+ such that km=n\displaystyle km=n.
If m\displaystyle m is a multiple of n\displaystyle n then jZ+\displaystyle \exists j\in\mathbb{Z}^+ such that jn=m\displaystyle jn=m.
Please finish the proof an post the result.
 
If n\displaystyle n is a multiple of m\displaystyle m then kZ+\displaystyle \exists k\in\mathbb{Z}^+ such that km=n\displaystyle km=n.
If m\displaystyle m is a multiple of n\displaystyle n then jZ+\displaystyle \exists j\in\mathbb{Z}^+ such that jn=m\displaystyle jn=m.
Please finish the proof an post the result.
I dont know how to.
 
I dont know how to.
Suppose that m is a multiple of n & n is a multiple of m\displaystyle m\text{ is a multiple of }n~\&~ n\text{ is a multiple of }m.
So KZ+[Kn=m] & JZ+[Jm=n]\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:
\(\displaystyle \begin{align*}m&=K\cdot n \\&=K\cdot(J\cdot m)\\&=(K\cdot J)\cdot m \\\large1 &= K\cdot J \end{align*}\)
Now what?\displaystyle \LARGE\text{Now what?}
 
If n\displaystyle n is a multiple of m\displaystyle m then kZ+\displaystyle \exists k\in\mathbb{Z}^+ such that km=n\displaystyle km=n.
If m\displaystyle m is a multiple of n\displaystyle n then jZ+\displaystyle \exists j\in\mathbb{Z}^+ such that jn=m\displaystyle jn=m.
Please finish the proof an post the result.
Thanks
Got it
 
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.
 
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