X - IDA.LiU.se
... • a – 1 is a multiple of every prime dividing m, i.e. a – 1 = k p , kN for every prime p such that m mod p = 0 • a – 1 is a multiple of 4, if m is a multiple of 4 ...
... • a – 1 is a multiple of every prime dividing m, i.e. a – 1 = k p , kN for every prime p such that m mod p = 0 • a – 1 is a multiple of 4, if m is a multiple of 4 ...
Integers, Prime Factorization, and More on Primes
... Proposition 7. Let a, b ∈ Z. Then a positive integer d is the greatest common divisor of a and b if and only if (1) d divides both a and b; (2) If c divides both a and b, then c divides d. Proof. If the above two conditions are satisfied by the integer d, it is clear that d is the largest one among ...
... Proposition 7. Let a, b ∈ Z. Then a positive integer d is the greatest common divisor of a and b if and only if (1) d divides both a and b; (2) If c divides both a and b, then c divides d. Proof. If the above two conditions are satisfied by the integer d, it is clear that d is the largest one among ...
A2 9-1 Notes Teacher Edition
... To find out when a rational expression is undefined, you have to find when the denominator equals 0. That means that you have to completely factor the original denominator. (Why?) Because the simplified denominator will miss (common) factors. ...
... To find out when a rational expression is undefined, you have to find when the denominator equals 0. That means that you have to completely factor the original denominator. (Why?) Because the simplified denominator will miss (common) factors. ...
![2 is irreducible in Q[ √ 2]](http://s1.studyres.com/store/data/016623007_1-1c8ad52cfddd987781a617fa5dd841ce-300x300.png)
Segment Addition Postulate
... The same number of hash marks show congruent segments (as shown to the right). 2. ______________ - the point that divides a segment into 2 congruent parts. ...
... The same number of hash marks show congruent segments (as shown to the right). 2. ______________ - the point that divides a segment into 2 congruent parts. ...
http://www.math.upenn.edu/~wilf/website/recounting.pdf
... 1. The denominator of each fraction is the numerator of the next one. That means that the nth rational number in the list looks like b(n)/b(n + 1) (n = 0, 1, 2, . . .), where b is a certain function of the nonnegative integers whose values are {b(n)}n≥0 = {1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4 ...
... 1. The denominator of each fraction is the numerator of the next one. That means that the nth rational number in the list looks like b(n)/b(n + 1) (n = 0, 1, 2, . . .), where b is a certain function of the nonnegative integers whose values are {b(n)}n≥0 = {1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4 ...
A Stirling Encounter with Harmonic Numbers - HMC Math
... different ways, yields both sides of the identity. Combinatorial proofs often provide intuitive and concrete explanations where algebraic proofs may not. For example n−1 ...
... different ways, yields both sides of the identity. Combinatorial proofs often provide intuitive and concrete explanations where algebraic proofs may not. For example n−1 ...
Addition
Addition (often signified by the plus symbol ""+"") is one of the four elementary, mathematical operations of arithmetic, with the others being subtraction, multiplication and division.The addition of two whole numbers is the total amount of those quantities combined. For example, in the picture on the right, there is a combination of three apples and two apples together; making a total of 5 apples. This observation is equivalent to the mathematical expression ""3 + 2 = 5"" i.e., ""3 add 2 is equal to 5"".Besides counting fruits, addition can also represent combining other physical objects. Using systematic generalizations, addition can also be defined on more abstract quantities, such as integers, rational numbers, real numbers and complex numbers and other abstract objects such as vectors and matrices.In arithmetic, rules for addition involving fractions and negative numbers have been devised amongst others. In algebra, addition is studied more abstractly.Addition has several important properties. It is commutative, meaning that order does not matter, and it is associative, meaning that when one adds more than two numbers, the order in which addition is performed does not matter (see Summation). Repeated addition of 1 is the same as counting; addition of 0 does not change a number. Addition also obeys predictable rules concerning related operations such as subtraction and multiplication.Performing addition is one of the simplest numerical tasks. Addition of very small numbers is accessible to toddlers; the most basic task, 1 + 1, can be performed by infants as young as five months and even some non-human animals. In primary education, students are taught to add numbers in the decimal system, starting with single digits and progressively tackling more difficult problems. Mechanical aids range from the ancient abacus to the modern computer, where research on the most efficient implementations of addition continues to this day.