In terms of division, we say that a divides b if and only if the remainder is zero when b is divided by a. In the late third century, the Chinese mathematician Sun Tzu asked his studen...In terms of division, we say that a divides b if and only if the remainder is zero when b is divided by a. In the late third century, the Chinese mathematician Sun Tzu asked his students: "We have things of which we do not know the number; if we count by threes, the remainder is 2; if we count be fives, the remainder is 3; if we count by sevens, the remainder is 2.
The notation “(mod n)” after m_1\equiv m_2 indicates a congruence relation, in which “mod n” are enclosed by a pair of parentheses, and the notation is placed at the end of the congruence....The notation “(mod n)” after m_1\equiv m_2 indicates a congruence relation, in which “mod n” are enclosed by a pair of parentheses, and the notation is placed at the end of the congruence. Given any integer m, m\bmod n \in\{0,1,2,\ldots,n-1\}. \nonumber We call these values the residues modulo . In modular arithmetic, when we say “reduced modulo ,” we mean whatever result we obtain, we divide it by n, and report only the smallest possible nonnegative residue.