In the above statement N , a , b are natural numbers . I was wondering whether the above statement is always true . If it is always true will anyone give me a simple reason or proof for it ? Please guide me .
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Yes its always true. Consider the fundamental theorem of arithmetic: n has a unique prime decomposition, as do $a$ and $b$. – David Diaz Jan 09 '20 at 16:14
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related and likely inspiration for this question: https://math.stackexchange.com/questions/3502984/if-a-polynomial-px-is-divisible-by-2-polynomials-x-a-and-x-b-is-it-necessa – gt6989b Jan 09 '20 at 16:19
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1If $ax+by=1$ then $Nax+bny=N$ and each term is divisible by $ab.$ – Thomas Andrews Jan 09 '20 at 16:23
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i.e. $\ ab\mid aN,bN,\Rightarrow,ab\mid (aN,bN) = (a,b)N = N\ $ in gcd or ideal language (cf. various forms of Euclid's Lemma) – Bill Dubuque Jan 10 '20 at 00:28
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well yes, since: a/N and b/N $\rightarrow$ $N = a* \alpha , N = b*\beta $
we have: $a/N \rightarrow a/b*\beta \rightarrow a/ \beta$ (because a and b are coprime by Gauss theorem)
Hence: $\beta = 0 or \beta = k*a \rightarrow N = a*b*k$
Therefore: a*b/N
Oussama Amir
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