July 7, 2009 Modular arithmetic is a branch of number theory
that is useful in its own right and as a tool in such
disciplines as integer factorization, calendrical and
astronomical calculations, and cryptography. Modular arithmetic
was systematized by Carl Friedrich Gauss in his book
Disquisitiones Arithmeticae, published in 1801. Modular
arithmetic is a system of arithmetic for integers in which all
results are expressed as non-negative integers less than the
modulus. A familiar use of modular arithmetic is the twelve-hour
clock, in which eight hours after nine o’clock is five o’clock;
that is, 9 + 8 ≡ 5 (mod 12). The ≡ sign used in modular
arithmetic specifies congruence, not equality. Two numbers are
congruent in a given modulus if the difference between them is
an even multiple of the modulus; for instance 17 ≡ 5 (mod 12).
Modular addition is like regular addition, but “wraps around” at
the modulus. Modular subtraction is the inverse operation of
modular addition, and modular multiplication is just repea