Divisibility in the Integers. Intro to Analysis

[Windowmaker]

Homework Statement

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

Assume a, b and c are integers.

Homework Equations

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 doesn't seem to get me anywhere. Thanks in advance.

 

[SammyS]

Homework Statement

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

Assume a, b and c are integers.

Homework Equations

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 doesn't seem to get me anywhere. Thanks in advance.

You have b in terms of a ? ...

 

[Windowmaker]

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?

 

[Jamin2112]

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]

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| ? .

 

[Windowmaker]

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

 

[SammyS]

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

Well, I saw that you were almost there in your original post. I'd rather lead you to discover what's missing than to simply provide the bridge.

Good luck with the proofs. It can be challenging, coming up with some of them, but very rewarding once you do !

 

[Windowmaker]

What sucks about this course is there's only like 10 problems per section. He collects all 10 or so problems, so there's no extra problems to work out. Its not overly difficult, its just a bit different than the math I am used too. Computational math is much different than proof based math. Lol.