gcd need help?

Question

d = gcd(a,b) ; a = da' and b = db', show that gcd (a', b') =1

Answer 1

If gcd(a',b') = e (<>1)

Then a' = ea'' and b' = eb''
Then a = dea'' and b = deb'' and
de>d so d would NOT be the gcd, unless e = 1.

Answer 2

Suppose a = 30 = 2*3*5, and
b = 42 = 2* 3* 7
Then d = gcd(30,42) = 3
a' = a/d =30/3 = 10, and
b' = b/d = 42/3 = 14
So gcd(a',b') = gcd(10,14) = 2
This is a counter example of your hypothesis, showing that it is not true.

Answer 3

by contradiction:
Assume 1 is not gcd(a',b')
let a' =a/d and b' = b/d
let p not equal to 1 be the gcd(a',b')
then a'=pa'' and b'=pb''
substitute:
a=dpa'' and b=dpb''
therefore dp= gcd(a,b) does not equal d.
So, d is and isn't the gcd(a,b)
therefore, by contradiction gcd(a',b')=1