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The remainder of a division of two integers $\frac {a}{b},\ b \neq 0$ is the integer $r < b$ such that $a = qb + r$, where $q$ is the quotient; in other words, $r$ is the part of $a$ that is not divisible by $b$. If $a = 4$, and $b = 3$, for example, the division $\frac {4}{3}$ would have remainder $1$, since $4 = (1)3 + 1$ (notice that the quotient, in this case, is one). If $b$ is a divisor of $a$, the remainder is said to be zero.

The concept of a remainder is related to modular arithmetic: $r$ is said to be the residue class of $a$ in modulo $b$ iff $a = qb + r$ (an equivalent statement would be $a \equiv r \mod b$).

It is important to notice that the remainder is most useful when an integer quotient is desired, as we can always say that $a = qb$ for any real number $q$ (in the example provided earlier, $q = 1.\overline{3}$).

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