Let i, j, d ∈N a t, with i>j and d>0. Prove that: d \(i-j) iff rem(i, d)=rem(j, d) .

Name: Problem 31, Chapter 3: Logic, sets, and recursion
Uploaded: 2020-07-14T20:35:52-08:00
Duration: 6 min 6 s
Channel: Michael Jacobsen
Description: Let i, j, d ∈N a t, with i>j and d>0. Prove that: d \(i-j) iff rem(i, d)=rem(j, d) .

Let i, j, d ∈N a t, with i>j and d>0. Prove that: d \(i-j) ...

Key Concepts

Modular Arithmetic

Modular arithmetic involves working with remainders after division by a positive integer (the modulus). It provides a framework where two integers are considered equivalent if they yield the same remainder when divided by the modulus. This equivalence relation, denoted as congruence modulo d, is fundamental to proving that two numbers have the same remainder if and only if their difference is divisible by d.

Euclidean Division Theorem

The Euclidean Division Theorem guarantees that for any integer and any positive divisor, there exist unique integers representing the quotient and the remainder. This theorem lays the groundwork for understanding modular arithmetic as it shows that every integer can be expressed as a multiple of the divisor plus a remainder, which is essential for connecting divisibility with congruence.

Divisibility

Divisibility is a core concept in number theory where an integer divides another if there exists an integer multiplier that gives the second number when multiplied by the divisor. In this context, the statement d divides (i - j) asserts the existence of such an integer that ties the difference of two numbers to the modulus d.

Transcript

00:01 In this video, we have a matrix a of size n by n and a matrix b of size n by p.

00:08 We're going to refer to all the entries of the first matrix a as a .i .j.

00:13 And for b, the entries will be referred to as b -i -j.

00:18 So what we're going to do here is first prove that r times the matrix product, a times b, is equal to r times a times b, and then, then also show that this is as well equal to a times r times b.

00:37 If we can prove this statement, we'll know that the scalar r can be arranged in any way we like in the sense that we have a certain kind of commutative law.

00:47 R can go either here, in front of a, or outside of the entire product, scaling the result.

00:53 So let's show how equality can be ascertained in both of these statements.

00:59 We're going to use the rule for matrix multiplication.

01:02 That says the following.

01:04 The ij entry of a times b, referred in this way, is equal to take a .i .1 times b .j .1.

01:17 Add to it, a .i .2 times b .j2.

01:23 And multiply all the way until we get to the entries a .i .n.

01:29 Times b .j .n.

01:30 That's a j there.

01:32 So that's our rule for multiplying a.

01:34 B so let's see what would happen if we consider this side the left -hand side of the equation we're trying to prove we'll have r or rather the ij entry of r times a times b is r times all of this result here that we just formed it'll be a i1 times b j 1 plus a i2 times b j2 all the way to until we get to a -i -n times b -j -n.

02:15 So that's the result of r times the matrix product, a times b.

02:21 Let's go now to the next product, where we first scale a by r and then multiply the result by b.

02:28 Here we have that the ij entry of r times a times the matrix b is of the following.

02:43 Well first consider what happens when we take r times a.

02:46 The effect is every entry of a referred to as a ij is multiplied by r...

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