Question

if a is a prime, a not divide b, how to prove gcd(a,b) =1?
Please, help

Answer 1

@ganeshie8

Answer 2

if b is a prime, then we are done
if b is not a prime, then b = p1p2....pn and none of them = a, then?

Answer 3

then a and b have no common factors

Answer 4

well except 1 (or -1) but you know gcd can only be positive or zero

Answer 5

:) they can be -1 also (in Ring)

Answer 6

Thank you very much.

Answer 7

I think what you said is very good.
\[ \text{ Assume } \text{ the prime factorization of } b \text{ is } p_1 \cdot p_2 \cdots p_n \\ \text{ that is } b=p_1 \cdot p_2 \cdots p_n \\ gcd(a,b)= \gcd(a,p_1 \cdot p_2 \cdot p_3 \cdots p_n)=1 \text{ since } a \neq p_i \text{ where } i=1,2,....,n \\ \text{ and we know } a \neq p_i \text{ since } a \text{ does not divide b} \text{ and } a \text{ is a prime } \]

Answer 8

I understand the concept just don't know how to put it in neat. :)

Answer 9

lol putting it in neat
i know what you mean
some people are way neater than me
but i think what we have above is pretty neat :p
but that is my opinion and it may differ with other people