Proof: Quotient and Remainder involving floor function

[whatlifeforme]

Homework Statement

Show that if a (is in) Z and d (is in) Z+, d>1 then the quotient and remainder when a is divided by d are a/d and a-d(floor function(a/d))

Homework Equations

The Attempt at a Solution

solution (that i have from handout - that i don't understand)

by thm 2 p202 (? i am not sure what it says as it is in handwriting, but it looks like p202) we know a = dq+r 0<=r<d dividing the equation by d we have a/d = q + r/d w/ 0<=r/d<1, hence by definition it's clear q is (floor function(a/d)) while the original equation shows r = a-dq giving the second result..

 

[mighty2000]

Hello,

Thank you for your post. I can help you understand the solution provided in the handout.

First, let's break down the notation used in the solution.

- The symbol "Z" represents the set of integers, so when it says "a (is in) Z", it means that a is an integer.
- The symbol "Z+" represents the set of positive integers, so when it says "d (is in) Z+, d>1", it means that d is a positive integer greater than 1.
- The symbol "a/d" represents the quotient when a is divided by d.
- The symbol "a-d(floor function(a/d))" represents the remainder when a is divided by d.

Now, let's look at the solution step by step.

1. The first step is to write the equation a = dq + r, where d is the divisor, q is the quotient, and r is the remainder. This is a standard form for division, where the remainder is always less than the divisor. In this case, we know that the remainder r is between 0 and d-1, because it is less than the divisor d.

2. Next, we divide both sides of the equation by d to get a/d = q + r/d. This is a standard form for expressing the quotient and remainder when dividing by d.

3. From the previous step, we can see that q is the integer part of the quotient a/d, and r/d is the fractional part. This is because r is between 0 and d-1, so r/d is between 0 and 1.

4. By definition, the floor function of a number is the largest integer less than or equal to that number. In other words, it rounds the number down to the nearest integer. So, in this case, the floor function of a/d is q.

5. Finally, we can substitute the value of q into the original equation a = dq + r to get a = d(floor function(a/d)) + r. This shows that the remainder when a is divided by d is a-d(floor function(a/d)).

I hope this explanation helps you understand the solution better. If you have any further questions, please let me know.