1. Feb 10, 2007

### kiss89

Hi, i'm taking calculus this sem, and i have a question on stuff pre calculus.

If x and y are natural numbers and y[less-than]x, then whole numbers q and r must exist such that x=yq+r.
a)what is the value of r if y is a factor of x? IT IS ZERO
b)If y is not a factor of x, what are the possible values of r if y=5 , y=7, or y=n???

I dont get b), can anyone help me plz.
THANK YOU.

2. Feb 10, 2007

### cristo

Staff Emeritus
Well, if y=5, then we can write x=5q+r (5<x). This says that x can be written as a multiple of 5 plus some remainder. What is the maximum value that this remainder can be?

3. Feb 10, 2007

### kiss89

there is no max. value, and by the way the answer at the back of the book is : 1,2,3,4(for y=5) and 1,2,3,4,5,6(for y=7) and 1,2,3,....,n-1( for y=n).
but i still dont understand what these answers mean.

4. Feb 10, 2007

### d_leet

Why do you think there is no max value?

5. Feb 10, 2007

### cristo

Staff Emeritus
Ok, so I suppose you could say there is no maximum value for r. However, suppose we say that we want x=5q+r where x>5, and q is as large as possible (which is what the question wants). i.e. we write x as the largest multiple of 5 possible, then take r as the remainder. Now, with the question written like this, there is a maximum value for r.

I'm not too sure how to give any futher hints without giving away the answer! But, suppose that r=9; is q the biggest value possible?

6. Feb 11, 2007

### HallsofIvy

It would help to state this properly: If x, y are natural numbers and y< x, then there exist unique whole numbers q and r such that x= yq+ r and $0\le r< y$

7. Feb 11, 2007

### mathwonk

as phrased, the correct answers are:
a) any multiple of y,
b) any integer at all.

(In this question, understanding how to ask it correctly is more important than answering it.]