## Main Question or Discussion Point

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.

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?

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.

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?

HallsofIvy
Homework Helper
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.
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 $0\le r< y$

a)what is the value of r if y is a factor of x? IT IS ZERO[\quote]
Without the additions above, x= 24, y= 4, you could write 24= 4(6) or 24= 4(5)+ 4 or 24= 4(4)+ 8, etc.

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.
If y= 5, then obviously the possible values for r are 1, 2, 3, 4. If y= 7, then 1, 2, 3, 4, 5, 6.

mathwonk