Expert: Sherry Wallin - 7/7/2010

Question
Sir,
can u pl explain me modulus function

Answer
ADITYA~
    The modulus function can be looked at in a number of ways. For the most part people view it as congruence where congruence acts like equals. Three horizontal bars is typically used for congruence (instead of two like in =) but since I don't have that symbol on my keyboard I will use the word 'cong' to mean congruent.

So if a cong b mod n then (a-b) = kn where k is an integer. This also means that both a and b will have the same remainder when being divided by n.
A concrete example would be 5 cong 3 mod 2 because 5/2 = 2*2 + r1 where here n = 2, k = 2, and the remainder is 1. 3/2 = 1*2 + 1 where n = 2, k = 1 and r = 1. Also note that 5-3 = 2 and 2/2 = 1.  Military time provides a good way to discuss modular arithmetic: 1400 hours is really 200pm because 1400 cong 200 mod 1200. 1400/1200 = 1*1200 + 200 and 200/1200 = 0*1200 + 200. When a mod is reduced it has a non negative value that is the smallest value that is non negative. For example 5 cong -9 mod 7 because 5/7 = 0*7 + 5 and -9/7 = -1*7 -2 but what happens if you add 7 to -2? Yes you get 5. Also 5 -(-9) = 14 which is divisible by 7 since 14 = 7*2. If the remainder is 0 then you have the definition of divides: we say a divides b and we write a|b and translate this to mean b = ka where k is some integer.

Math Prof