Question

Find the order of $\frac{\mathbf{8}}{\mathbf{9}}\mathbf{,}\frac{\mathbf{14}}{\mathbf{5}}$ and$\frac{\mathbf{48}}{\mathbf{28}}$ in the additive group $\mathrm{\mathbb{Q}}/\mathrm{\mathbb{Z}}$.

Short answer

The order of $\frac{8}{9}$Is 9.

The order of $\frac{14}{5}$is 5.

The order of $\frac{48}{28}$is 7.

Step-by-step solution

Finding order of 89 in ℚ/ℤ

Consider n be an arbitrary element such that$\mathrm{n}\in \mathrm{\mathbb{Z}}$is the order of any element in the additive group $\mathrm{\mathbb{Q}}/\mathrm{\mathbb{Z}}$.

Therefore, we can also write that $\mathrm{n}\left(\frac{8}{9}\right)\in \mathrm{\mathbb{Z}}$

So, n is the smallest number for which $n\left(\frac{8}{9}\right)$must be an integer. In this case, 9 is the smallest integer such that$9\left(\frac{8}{9}\right)=8$

Therefore, order of $\frac{8}{9}$is 9

Finding order of 145 in ℚ/ℤ

Consider n be an arbitrary element such that$\mathrm{n}\in \mathrm{\mathbb{Z}}$is the order of any element in the additive group $\mathrm{\mathbb{Q}}/\mathrm{\mathbb{Z}}$.

Therefore, we can also write that $n\left(\frac{14}{5}\right)\in \mathbb{Z}$

So, n is the smallest number for which $n\left(\frac{14}{5}\right)$must be an integer. In this case, 5

is the smallest integer such that $5\left(\frac{14}{5}\right)=14$.

Therefore, order of$\frac{14}{5}$ is 5.

Finding order of 4828 in ℚ/ℤ

Consider n be an arbitrary element such that $\mathrm{n}\in \mathrm{\mathbb{Z}}$ is the order of any element in the additive group $\mathrm{\mathbb{Q}}/\mathrm{\mathbb{Z}}$.

Therefore, we can also write that $\mathrm{n}\left(\frac{48}{28}\right)\in \mathrm{\mathbb{Z}}$

So, n is the smallest number for which $n\left(\frac{48}{28}\right)$must be an integer. In this case, 13 is the smallest integer such that $7\left(\frac{48}{28}\right)=12$.

Therefore,order of$\frac{48}{28}$ is 7.