Discrete Math CS 2800
... If a = 7 and d = 3, then q = 2 and r = 1, since 7 = (2)(3) + 1. If a = −7 and d = 3, then q = −3 and r = 2, since −7 = (−3)(3) + 2. So: given positive a and (positive) d, in order to get r we repeatedly subtract d from a, as many times as needed so that what remains, r, is less than d. Given negati ...
... If a = 7 and d = 3, then q = 2 and r = 1, since 7 = (2)(3) + 1. If a = −7 and d = 3, then q = −3 and r = 2, since −7 = (−3)(3) + 2. So: given positive a and (positive) d, in order to get r we repeatedly subtract d from a, as many times as needed so that what remains, r, is less than d. Given negati ...
4 slides/page
... The equation ax = b for a, b ∈ R is uniquely solvable if a 6= 0: x = ba−1. • Can we also (uniquely) solve ax ≡ b (mod m)? • If x0 is a solution, then so is x0 + km ∀k ∈ Z ◦ . . . since km ≡ 0 (mod m). So, uniqueness can only be mod m. But even mod m, there can be more than one solution: • Consider 2 ...
... The equation ax = b for a, b ∈ R is uniquely solvable if a 6= 0: x = ba−1. • Can we also (uniquely) solve ax ≡ b (mod m)? • If x0 is a solution, then so is x0 + km ∀k ∈ Z ◦ . . . since km ≡ 0 (mod m). So, uniqueness can only be mod m. But even mod m, there can be more than one solution: • Consider 2 ...
Full text
... not divisible hyp is 2*13*24*3... rSr~l, where % is the number of I'S in the base p expansion of n. Proof: First, we note that the maximum exists. It is well known that rx
... not divisible hyp is 2*13*24*3... rSr~l, where % is the number of I'S in the base p expansion of n. Proof: First, we note that the maximum exists. It is well known that rx
1, so r < p +1. By Kummer's Theorem for Generalized Binomial Coefficients, /?|[£] g if ...
List of important publications in mathematics
This is a list of important publications in mathematics, organized by field.Some reasons why a particular publication might be regarded as important:Topic creator – A publication that created a new topicBreakthrough – A publication that changed scientific knowledge significantlyInfluence – A publication which has significantly influenced the world or has had a massive impact on the teaching of mathematics. Among published compilations of important publications in mathematics are Landmark writings in Western mathematics 1640–1940 by Ivor Grattan-Guinness and A Source Book in Mathematics by David Eugene Smith.