So I was just messing around on my calculator and I noticed that for every positive integer the formula $\frac{n-1}{n}$ is always a fraction in its simplest form im not quite sure why this happens is there a case where this is not true?
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1Any common factor of $n$ and $n-1$ divides $1$. – Kavi Rama Murthy Feb 02 '21 at 09:10
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When $n=1$ you get a potential exception. – Mark Bennet Feb 02 '21 at 09:14
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@mark Bennet Its still in its simplest form though – The homeschooler Feb 02 '21 at 09:18
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I don't think $\frac 01$ would be considered by most people the simplest form for $0$? – Mark Bennet Feb 02 '21 at 09:29
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Using Bezout's Lemma, we know that $\text{gcd}(n,n-1)=1$ because $n-(n-1)=1$. This means they have no common prime factors, so you can't find any natural number $k\neq 1$ such that $k\mid n$ and $k\mid(n-1)$, hence the fraction $\frac{n-1}{n}$ is always in its simplest form.
Alejandro Bergasa Alonso
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Because $n-(n-1)=1$ and this means that $n$ and $n-1$ are co-prime. Can you see why ?
Any common divisor to $n$ and $n-1$ divides their difference...
marwalix
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