Euclidean division ($a=bq+r,0\le r<b$)

For an integer $a$ and positive integer $b$, Euclidean division writes $a=bq+r$ with unique integer quotient $q$ and remainder $0\le r<b$. For $a\ge 0$, repeated quotient-and-remainder steps yield the digits of a base-p expansion when $b=p$. The uniqueness follows because the difference of two permitted remainders cannot be a nonzero multiple of $b$.

For an integer $a$ and positive integer $b$, the remainder in Euclidean division is the unique integer $r$ in $0\le r<b$ for which $a=bq+r$ with an integer $q$. It is not the full quotient $a/b$.

Repeated division with remainder that determines successive digits of a quotient. To generate the fractional digits in base $b$, multiply the current remainder by $b$, divide by the denominator, and keep the resulting remainder for the next step.