Question

Is `1/42` a repeating or terminating decimal?

Answer 1

It repeats:

`1/42 = 0.0bar(238095)`

Explanation:

Note that:

`42 = 2 * 3 * 7`

has factors which are not factors of `10`.

As a result, the decimal expansion of `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)`
`color(white)(0000")")color(white)(00".")1color(white)(0)6color(white)(0)0`
`color(white)(0000")")color(white)(00".")underline(1color(white)(0)2color(white)(0)6_`
`color(white)(0000")")color(white)(00"."00)3color(white)(0)4color(white)(0)0`
`color(white)(0000")")color(white)(00"."00)underline(3color(white)(0)3color(white)(0)6)`
`color(white)(0000")")color(white)(00"."000000)4color(white)(0)0color(white)(0)0`
`color(white)(0000")")color(white)(00"."000000)underline(3color(white)(0)7color(white)(0)8)`
`color(white)(0000")")color(white)(00"."00000000)2color(white)(0)2color(white)(0)0`
`color(white)(0000")")color(white)(00"."00000000)underline(2color(white)(0)1color(white)(0)0)`
`color(white)(0000")")color(white)(00"."0000000000)1color(white)(0)0`

So:

`1/42 = 0.0bar(238095)`