Find the decimal expansions of $\frac{10}{3},\, \frac{7}{8}$ and $\frac{1}{7}$.

$\frac {10}{3}= 3.3333$ ............ 

Remainders : $1$, $1$, $1$, $1$, $1$ ............

Divisor : $3$

$\frac {7}{8}= 0.875$ ............ 

Remainders : $6$, $4$, $0$, ............

Divisor : $8$

$\frac {1}{7}= 0.142857$ ............ 

Remainders : $3$, $2$, $6$, $4$, $5$, $1$, $3$, $2$, $6$, $4$, $5$, $1$ ............

Divisor : $7$

What have you noticed ? You should have noticed at least three things: 

$(i)$ The remainders either become $0$ after a certain stage, or start repeating themselves.

$(ii)$ The number of entries in the repeating string of remainders is less than the divisor (in $\frac {10 }{3}$ one number repeats itself and the divisor is $3$, in $\frac {1 }{7}$ there are six entries $326451$ in the repeating string of remainders and $7$ is the divisor). 

$(iii)$ If the remainders repeat, then we get a repeating block of digits in the quotient (for $\frac {10} {3}$ , $3$ repeats in the quotient and for $\frac {1} {7}$ , we get the repeating block $142857$ in the quotient).