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In this case, s is called the least absolute remainder. [3] As with the quotient and remainder, k and s are uniquely determined, except in the case where d = 2n and s = ± n. For this exception, we have: a = k⋅d + n = (k + 1)d − n. A unique remainder can be obtained in this case by some convention—such as always taking the positive value ...
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.
Modular exponentiation is the remainder when an integer b (the base) is raised to the power e (the exponent), and divided by a positive integer m (the modulus); that is, c = b e mod m. From the definition of division, it follows that 0 ≤ c < m. For example, given b = 5, e = 3 and m = 13, dividing 5 3 = 125 by 13 leaves a remainder of c = 8.
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 ] For example, the expression "5 mod 2" evaluates to 1, because 5 divided by 2 has a quotient of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0 ...
where denote a specific function from the domain to the natural numbers called a "Euclidean function". The uniqueness of the quotient and the remainder remains true for polynomials, but it is false in general.
President-elect Donald Trump complained on Friday that American flags would still be lowered to half-staff in honor of the late President Jimmy Carter during Trump's Jan. 20 inauguration.
Dividing a(x) by b(x) yields a remainder r 0 (x) = x 3 + (2/3)x 2 + (5/3)x − (2/3). In the next step, b ( x ) is divided by r 0 ( x ) yielding a remainder r 1 ( x ) = x 2 + x + 2 . Finally, dividing r 0 ( x ) by r 1 ( x ) yields a zero remainder, indicating that r 1 ( x ) is the greatest common divisor polynomial of a ( x ) and b ( x ...
Environmentalist Ellen Swallow Richards was the first woman admitted to the Massachusetts Institute of Technology, an impressive feat in and of itself.What's even more admirable was her work in science, a field in which women faced many obstacles, as well as the time she spent getting her Ph.D. in chemistry from MIT– well, almost.