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

Question

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 .

Answer 1

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