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0. Modulo Method First need to divide the Dividend by the Divisor: 2 4 = 0.50 Next we take the Whole part of the Quotient (0) and multiply that by the Divisor (4): 0 x 4 = 0. And finally, we take the answer in the second step and subtract it from the Dividend to get the answer to 2 mod 4: 2 - 0 = 2.
Definition. The Modulus is the remainder of the euclidean division of one number by another. % is called the modulo operation. For instance, 9 divided by 4 equals 2 but it remains 1. Here, 9 / 4 = 2 and 9 % 4 = 1. In your example: 5 divided by 7 gives 0 but it remains 5 (5 % 7 == 5). Calculation.
Similar trick can be done for integer division by 10 where we note that we always can write a number x as x = a 10 + b, where a = x/10 and b = x%10, then again we can study x1*x2 and x1+x2 to deduce similar rules for the integer division of 128 bit integers taking advantage of the fast versions of the 64 bit ones.
The basic outline of the optimisation of modulus looks like this: If you had exact arithmetic, you could replace x % p with p * ((x * (1/p)) % 1). For constant p, 1/p can be precomputed at compile time. The %1 operation simply consists of discarding the fraction part, which is just a right-shift. So that replaces a division with two multiplies ...
The difference between mod and rem is subtle, but important. (-1 mod 2) would normally give 1. More specifically given two integers, X and Y, the operation (X mod Y) tends to return a value in the range [0, Y). Said differently, the modulus of X and Y is always greater than or equal to zero, and less than Y.
81. Yes, those functions act differently. As defined in the official documentation: quot is integer division truncated toward zero. rem is integer remainder, satisfying: (x `quot` y)*y + (x `rem` y) == x. div is integer division truncated toward negative infinity. mod is integer modulus, satisfying: (x `div` y)*y + (x `mod` y) == x.
In modular arithmetic, one defines classes of numbers based on the modulo. In other words, in modulo m arithmetic, a number n is equivalent (read: the same) to n + m, n - m, n + 2m, n - 2m, etc. One defines m "baskets" and every number falls in one (and only one) of them. Example: one can say "It's 4:30 pm" or one can say "It's 16:30".
Modulus is just the remainder in division before its used in a decimal quotient. Example: The division of two numbers is often expressed as a decimal number (quotient). But the result of the division of say, 1/3, can also be expressed in whole numbers as "0 with a remainder of 1". But that form of quotient is not very helpful in modern math, so ...
4. Answering your specific questions: mod is a remainder operator. If applied to a series of numbers x in 0, 1, ..., then x mod n will be 0, 1, ..., n-1, 0, 1, ..., n-1, ad infinitum. When your modulus n is a power of 2, then x mod n will count up in binary from 0 to n-1, back to 0, to n-1, etc; for modulus n that looks like binary 01xxxxx, x ...
mod = a % b. This stores the result of a mod b in the variable mod. And you are right, 15 mod 4 is 3, which is exactly what python returns: >>> 15 % 4. 3. a %= b is also valid. edited Jul 28, 2019 at 16:12. wjandrea.