Algebra Gymnasiematematik(high school math)

Modular arithmetic & the Chinese Rest Theorem

modular arithmetic (sometimes called clock arithmetic) is a system of arithmetic for integers, where numbers ”wrap around” upon reaching a certain value—the modulus.

An example of this is the clock which starts repeating itself when it has reached 12. Therefore it is said that the clock follows an arithmetic modulo 12.  

Ex: If you add 7 hours to 8 o’clock the result is not 15 since when thee clock reaches zero it starts over from zero once again so the result is 7+8=3 in this arithmetic.

For a positive integer n, two integers a and b are said to be congruent modulo n,
a=b mod(n)
if the the remainder is the same when a and b are divided by n.

if their difference a − b is an integer multiple of n. The number n is called the modulus of the congruence.

a \equiv b \pmod n,\, 
e.g. 14

14≡12 (mod 2)

An example of an application of this mathematical tool is the Chinese Remainder Theorem.