Question

Show that for nonzero integers m and n, g.c.d (m,n) is the largest natural number dividing m and n.
Please help.

Answer 1

thats precicely the definition of gcd

are you really after proving a definition ?

Answer 2

I am new in the field, so that I am not familiar with the terminology. I need step by step how to solve it. Please.

Answer 3

Is this a homework problem ?

Answer 4

yes

Answer 5

what definitions do we get to use
(because I actually use that as a definition above)

Answer 6

I don't know what "the largest natural number dividing m and n" mean

Answer 7

oh
pretend we have m=dk and n=dl
if d is the largest dividing number of m and n
then gcd(m,n)=gcd(dk,dl)=d if gcd(k,l)=1

Answer 8

Obviously, if m,n are not relative prime, it has a gcd

Answer 9

just nitpicking on your earlier statement :
```
Obviously, if m,n are not relative prime, it has a gcd
```

if m,n are relatively prime, the gcd is `1`
we never say "gcd is not there"

Answer 10

and by definition of gcd, we can find it out by the steps like
let m = an + a1 where 0<= a1<a
and so on....
I know how to go the steps, but feel not good if the question is just about "memorize" the step how to prove the definition

Answer 11

do you want to know how to find gcd given two numbers ?

Answer 12

I know how to find it,@rational by both ways, arithmetic and matrix.

Answer 13

then whats the question exactly ?

Answer 14

Let gcd(m,n)=d.
Then this means d|m and d|n.
This means dk=m and da=n
for integers k and a.
So gcd(m,n)=d only iff gcd(k,a)=1.

Like this is the only way I can better explain the definition above.

Answer 15

And I don't mean to be repetitive. This trying to state it better.

Answer 16

oops, hihihi... I am so sorry. It's not the correct problem on my assignment.
But I would like to know how to do it.
Again, I am sorry.

Answer 17

@myininaya I see !! thank you :)

Answer 18

Is this a definition in your class? (if so, you can't prove definitions.)
And if it isn't, what definition are you using for gcd?

Answer 19

No, it is a theorem.

Answer 20

Let me close this post and ... ask a new one. Please, be there to help