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If $\,\mathrm{gcd}(a,b)=1,\;d=ac,\;$ and $\,b\mid d,\,$ then $\,b\mid c.$

For example, $45=9\times 5$ is divisible by $3$ and $\mathrm{gcd}(3,5)=1$ and $3$ divides $9$. But it is easily noticeable that it just represents the natural property of a number and its multiple.

How can I proof it with mathematical formal way i.e using any sort of notation.

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  • Since $\gcd{(a,b)}=1$, you can always find integers $m$ and $n$ such that $$m \cdot a + n \cdot b=1$$ From the above you have $$m \cdot (ac) + n \cdot bc =c \qquad \cdots (1)$$

  • Use $d= ac$ and $b \mid d$ we have $d=k_{1}b$ in the above equation to get the answer.

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