Search results
Results from the WOW.Com Content Network
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, called the modulus of the operation. Given two positive numbers a and n, a modulo n (often abbreviated as a mod n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor. [1]
Consequently, (a mod m) denotes generally the unique integer k such that 0 ≤ k < m and k ≡ a (mod m); it is called the residue of a modulo m. In particular, (a mod m) = (b mod m) is equivalent to a ≡ b (mod m), and this explains why "=" is often used instead of "≡" in this context.
Modulo is a mathematical jargon that was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801. [3] Given the integers a, b and n, the expression "a ≡ b (mod n)", pronounced "a is congruent to b modulo n", means that a − b is an integer multiple of n, or equivalently, a and b both share the same remainder when divided by n.
Pascal chooses the result of the mod operation positive, but does not allow d to be negative or zero (so, a = (a div d) × d + a mod d is not always valid). [4] C99 chooses the remainder with the same sign as the dividend a. [5] (Before C99, the C language allowed other choices.)
c = b e mod m = d −e mod m, where e < 0 and b ⋅ d ≡ 1 (mod m). Modular exponentiation is efficient to compute, even for very large integers. On the other hand, computing the modular discrete logarithm – that is, finding the exponent e when given b , c , and m – is believed to be difficult.
A common technique used for division is called long division. Other methods include short division and chunking. [72] Integer arithmetic is not closed under division. This means that when dividing one integer by another integer, the result is not always an integer. For instance, 7 divided by 2 is not a whole number but 3.5. [73]
Another major difference between Division I and Division II is the physicality. That's not to say that there aren't strong athletes at the D2 level, but it's common to see bigger and faster ...
Long division is the standard algorithm used for pen-and-paper division of multi-digit numbers expressed in decimal notation. It shifts gradually from the left to the right end of the dividend, subtracting the largest possible multiple of the divisor (at the digit level) at each stage; the multiples then become the digits of the quotient, and the final difference is then the remainder.