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The definition and properties of divisibility with proofs of several properties. Formulas for quotient and remainder, leading into modular arithmetic.Video ...
We visit the idea of divisibility which you likely saw in grade school but give a formal definition. We use this to prove a common divisibility theorem. 0:00...
Please see the updated video at https://youtu.be/Qzy6hHgyb1gThe full playlist for Discrete Math I (Rosen, Discrete Mathematics and Its Applications, 7e) can ...
Definition (divisible) If a and b are integers, with a ≠ 0, we say a divides b if there exists an integer q such that b = a q. When a divides b we write a | b. When a does not divide b we write a ∤ b. From these definition we get special names for a, b, q. When we have b = a q, a is the divisor or factor of b. b is the dividend or multiple of a.
Divisibility 4.1 The division algorithm For the next few lectures we will exercise our ability to prove mathematical state-ments, using the fertile ground of number theory. In the process we will learn new proof techniques and tricks of trade. The number-theoretic concepts and results
Definition 4.1.1. Given a, b ∈ ℤ a,b\in\mathbb {Z}, we say that a a divides b b if there exists q ∈ ℤ q\in\mathbb {Z} such that b = q a b=qa. If this is the case we write a | b a|b and say that a a is a factor (or divisor) of b b, while b b is a multiple of a a ; otherwise we write a | b a\!\!\!\!\;\not\!\!\!\;|\!\;b. Example 4.1.2. (i)
Divisibility Rule for 3 and 9. A number is divisible by 3 or 9 if and only if the sum of its digits is divisible by 3 or 9, respectively. Note that this does not work for higher powers of 3. For instance, the sum of the digits of 1899 is divisible by 27, but 1899 is not itself divisible by 27. Proof.
