Question

Need help writing a simple proof for:
If d|a and a|b then d|b

Answer 1

Sure... first, what does it mean, exactly, if p|q ?

Answer 2

p divides q

Answer 3

yeah, so... what does it mean (in something we can use)

Answer 4

I don't know. this is the first proof I've ever done

Answer 5

if it were say 5|10 then q must be a multiple of p

Answer 6

That's right... q is a multiple of p :)
Or, formally, there exists an integer k such that
q = pk

okay?

Answer 7

Hey, I need to make sure you get this, this is the bread and butter of the proof... the very definition of p|k

Answer 8

rather p|q (not that it matters)

Answer 9

Ah ok, that part makes sense now.

Answer 10

I'm kinda slow, just starting school again

Answer 11

Okay, never lose sight of what we want to show, we want to show that
d|b
or in other words, there exists an integer k such that
b = dk

okay?

Answer 12

ok

Answer 13

Now, what are we given..
We know that
d|a
right?
So what can we immediately conclude?

Answer 14

a = dk?

Answer 15

Sure, but let's use m as our integer instead :)
there exists an integer m such that
a = dm.
Right?

Answer 16

ok

Answer 17

Now, we're also given a|b
then we can conclude? (use n this time, if you don't mind)

Answer 18

b = dn

Answer 19

no...
It's a|b
not d|b

Answer 20

d|b is what we want to prove.

Answer 21

so b = an

Answer 22

That's right :)
There exists an integer n such that b = an
But what is a equal to?

Answer 23

so we have b = dk, a = dm and b = an

Answer 24

We don't have b = dk !!!!

Answer 25

THAT IS WHAT WE WANT TO SHOW

Answer 26

I told you not to lose sight of what we want to prove.

Answer 27

oh right

Answer 28

If we already have b = dk, then there'd be no point of proving it.

Answer 29

lol

Answer 30

-.-

Answer 31

sorry

Answer 32

so we have a = dm and b = an

Answer 33

Okay, we have
a = dm
and
b = an

In particular, what is a equal to?

Answer 34

b/n

Answer 35

lol... okay, but use the first equation...
a = dm

Answer 36

(You're lacking creativity, that's rather crucial when it comes to proving)

Answer 37

Okay,
a = dm
and
b = an

We can replace the a in
b = an
with
dm
since a = dn


We get
b = dmn

Answer 38

Understood?

Answer 39

ohh

Answer 40

since m and n are both integers, then mn (their product) must also be an integer.

Answer 41

oh I see

Answer 42

So, if we let k = mn
then we get

b = dk
where k = mn is an integer...

QED
(Quod erat demonstrandum)
[Latin for "that which had to be demonstrated"]

Answer 43

so that is also a multiple

Answer 44

QED is what you put at the end of a proof
It's fancy :)

Answer 45

because it is just multiplied by an int

Answer 46

Yes.
So, everything understood?

Answer 47

yeah thank you so much :D

Answer 48

QED
Quite Easily Done

Answer 49

My brain is going to need to learn to think in new ways for this class

Answer 50

I'm used to calc :O

Answer 51

Proving needs creativity and some 'thinking outside the box'
Calculus is rather methodical.

Answer 52

yeah exactly

Answer 53

Welcome to the world of Fundamental Maths :)

Answer 54

Signing off now
----------------------------------
Terence out

Answer 55

Alrighty thanks again