Prove that ababat for every t is element of integerSolutionP

Prove that (a,b)=(a,b+at) for every t is element of integer.

PROOF : We know that

For every pair of whole numbers a and b there are two integers s and t such that as + bt = gcd(a, b).

also if we take s as a prime number then the above result becomes a+bt = gcd(a,b)

now (a,b)=(a,b+at) for every t is element of integer.

Then first we divide the bigger one by the smaller one:

33=1×27+633=1×27+6

Thus gcd(33,27)=gcd(27,6)gcd(33,27)=gcd(27,6). Repeating this trick:

27=4×6+327=4×6+3

and we see gcd(27,6)=gcd(6,3)gcd(27,6)=gcd(6,3).

at last 6=2×3+06=2×3+0

So since 6 is a perfect multiple of 3,

gcd(6,3)=3gcd(6,3)=3, and we have found that gcd(33,27)=3gcd(33,27)=3.

HENCE b i.e. bigger no. can be written as b=b+at

so the proof .