# If $a|N$ and $b|N$ where $a,b$ are coprime , is it necessary that $(a \times b) |N$? [duplicate]

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 .

• Yes its always true. Consider the fundamental theorem of arithmetic: n has a unique prime decomposition, as do $a$ and $b$. Jan 9, 2020 at 16:14
• related and likely inspiration for this question: math.stackexchange.com/questions/3502984/… Jan 9, 2020 at 16:19
• If $ax+by=1$ then $Nax+bny=N$ and each term is divisible by $ab.$ Jan 9, 2020 at 16:23
• 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) Jan 10, 2020 at 0:28

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: $$\beta = 0 or \beta = k*a \rightarrow N = a*b*k$$