Question

What is the 217th digit after the decimal point in the repeating decimal 0.3456?

Answer 1

`3`

Explanation:

Given:

`0.bar(3456)`

Notice that for any non-negative integer `n`, we have:

• The `4n+1`st digit after the decimal point is `3`
• The `4n+2`nd digit after the decimal point is `4`
• The `4n+3`rd digit after the decimal point is `5`
• The `4n+4`th digit after the decimal point is `6`

Note also that:

`217/4 = 54" "` with remainder `1`

That is:

`217 = 4*54+1`

So the `217`th digit after the decimal point is one of the `4n+1`st digits (with `n=54`) and must be `3`.