# Is 1/42 a repeating or terminating decimal?

May 3, 2018

It repeats:

$\frac{1}{42} = 0.0 \overline{238095}$

#### Explanation:

Note that:

$42 = 2 \cdot 3 \cdot 7$

has factors which are not factors of $10$.

As a result, the decimal expansion of $\frac{1}{42}$ does not terminate. It is a repeating decimal that we can find using long division.

Once the remainder in the long division repeats, so will the quotient.

color(white)(0000")")underline(color(white)(0)0"."0color(white)(0)2color(white)(0)3color(white)(0)8color(white)(0)0color(white)(0)9color(white)(0)5
4color(white)(0)2color(white)(0))color(white)(0)1"."0color(white)(0)0color(white)(0)0color(white)(0)0color(white)(0)0color(white)(0)0color(white)(0)0
color(white)(0000")")color(white)(0)underline(color(white)(0".")8color(white)(0)4)
$\textcolor{w h i t e}{0000 \text{)")color(white)(00".}} 1 \textcolor{w h i t e}{0} 6 \textcolor{w h i t e}{0} 0$
color(white)(0000")")color(white)(00".")underline(1color(white)(0)2color(white)(0)6_
$\textcolor{w h i t e}{0000 \text{)")color(white)(00".} 00} 3 \textcolor{w h i t e}{0} 4 \textcolor{w h i t e}{0} 0$
$\textcolor{w h i t e}{0000 \text{)")color(white)(00".} 00} \underline{3 \textcolor{w h i t e}{0} 3 \textcolor{w h i t e}{0} 6}$
$\textcolor{w h i t e}{0000 \text{)")color(white)(00".} 000000} 4 \textcolor{w h i t e}{0} 0 \textcolor{w h i t e}{0} 0$
$\textcolor{w h i t e}{0000 \text{)")color(white)(00".} 000000} \underline{3 \textcolor{w h i t e}{0} 7 \textcolor{w h i t e}{0} 8}$
$\textcolor{w h i t e}{0000 \text{)")color(white)(00".} 00000000} 2 \textcolor{w h i t e}{0} 2 \textcolor{w h i t e}{0} 0$
$\textcolor{w h i t e}{0000 \text{)")color(white)(00".} 00000000} \underline{2 \textcolor{w h i t e}{0} 1 \textcolor{w h i t e}{0} 0}$
$\textcolor{w h i t e}{0000 \text{)")color(white)(00".} 0000000000} 1 \textcolor{w h i t e}{0} 0$

So:

$\frac{1}{42} = 0.0 \overline{238095}$