# Divisibility in the Integers. Intro to Analysis

## Homework Statement

Prove: If a|b and b|c then a|c.

Assume a, b and c are integers.

none

## The Attempt at a Solution

If a divides b then that means that there is a

real integer "r" that is ra=b .

and since we assume b divides c then c=bs.

After here I got stuck. I was thinking maybe subsitute b and c for ar and bs, but it doesnt seem to get me anywhere. Thanks in advance.

SammyS
Staff Emeritus
Homework Helper
Gold Member

## Homework Statement

Prove: If a|b and b|c then a|c.

Assume a, b and c are integers.

none

## The Attempt at a Solution

If a divides b then that means that there is a

real integer "r" that is ra=b .

and since we assume b divides c then c=bs.

After here I got stuck. I was thinking maybe substitute b and c for ar and bs, but it doesnt seem to get me anywhere. Thanks in advance.
You have b in terms of a ? ...

Well I was thinking like we had 3|15. So a would be 3 and b would be 15. The r would be 5, so ra=b= 5*3=15. Or am I thinking about this wrong?

If a|b, then there exists an int r such that a*r = b.

If b|c, then there exists an int s such that s*b = c.

Since b = a*r, we have s*(a*r)=a*(r*s) = c.

Since r*s is an integer, this shows that c equals a multiplied by an integer, meaning a|c.

SammyS
Staff Emeritus
Homework Helper
Gold Member
Well I was thinking like we had 3|15. So a would be 3 and b would be 15. The r would be 5, so ra=b= 5*3=15. Or am I thinking about this wrong?
If this is a concrete example, you'll need another number.

You have 3|15, that's like a|b. Now you need b|c, so in the example, 15| ? .

Oh my goodness....First class on proofs its just simple subistution. Thanks man, cleared up alot!

SammyS
Staff Emeritus