Question

Let ${\mathit{m}}_{\mathbf{1}}\mathbf{,}{\mathit{m}}_{\mathbf{2}}\mathbf{,}\mathbf{\dots }\mathbf{\dots }{\mathit{m}}_{\mathbf{r}}$are pairwise relatively prime positive integers( that is$\mathbf{(}{\mathit{m}}_{\mathbf{i}}\mathbf{,}{\mathit{m}}_{\mathbf{j}}\mathbf{)}\mathbf{=}\mathbf{1}$, when $\mathit{i}\mathbf{\ne }\mathit{j}$),assume that${\mathit{m}}_{\mathbf{i}}\mathbf{/}\mathit{d}$ for each .Prove that${\mathit{m}}_{\mathbf{1}}{\mathit{m}}_{\mathbf{2}\mathbf{\dots }\mathbf{.}\mathbf{.}}{\mathit{m}}_{\mathbf{r}}\mathbf{/}\mathit{d}$

Short answer

It is proven that:$m_1{m}_2{m}_3\mathrm{.....}{m}_r|d$ .

Step-by-step solution

Determine the relatively prime integers

Assuming that:

$m_1{m}_2|d$are relatively prime positive integers pairwise.

Also, for${\text{m}}_{\text{i}}\text{|d}$each value of$\text{i}$.

Now, with the help of exercise six:$m_1{m}_2|d$.

Then, with the help of exercise five the values$m_1,m_2$and$m_3$are said to be relatively prime integers.

Proving the statement

It then gives:$m_1{m}_2|d$and$m_3|d$.

To conclude exercise six is again applied and then getting:${\text{m}}_{\text{1}}{\text{m}}_{\text{2}}{\text{m}}_{\text{3}}\text{|d}$.

This step is then continued till:$m_1{m}_2{m}_3\mathrm{.....}{m}_r|d$.

Therefore, it is proved:$m_1{m}_2{m}_3\mathrm{.....}{m}_r|d$ .