Learn / Library / Mental math

Far digit of a repeating decimal

Mental mathHard

The decimal expansion of $\frac{1}{7}$ repeats forever. Which digit sits in the $100$ th place after the decimal point?

Checking an answer needs JavaScript. The hint and the solution open without it.

Hint

Find the period, then reduce the position modulo the period.

Solution

$\frac{1}{7} = 0.142857142857 \dots$ , so the block $142857$ has period $6$ .
$100 = 6 \times 16 + 4$ , so position $100$ is the $4$ th digit of the block.
The block read left to right is $1$ , $4$ , $2$ , $8$ , $5$ , $7$ , so its $4$ th digit is $8$ .
The digit is $8$ .

Answer: $8$