please help me with this questionSolutionProof Say x and y

please help me with this question

Solution

Proof: -

Say x and y such that

x=l*m*n

and y=l*p*q as factors given l is gcd

then no other no. is common in there prime factorisation

since they have nothing in common so gcd =1

Exam1ple : -

We know gcd(6,9)=3

6=3*2 and 9=3*3

right?

so greatest common multiplier is 3

which means gcd is 3

so if we divide 3(i.e gcd) to both numbers we get 2 numbers which have nothing in common

eg, divide 6 and 9 by their gcd we get 2 and 3 respectively

and gcd of 2 and 3 is 1. So if two numbers are m,n have gcd d then definitely m=d*k and n=d*r for some random numbers k and r which are relatively prime(because if some other is also common then d will not be the gcd)

gcd(m/d,n/d)=gcd(k,r)