Thread: Associates

Hello all,

My book gave a theorem that stated that if a and a' both divide x and y, and that every g that divides x and y also divides a and a', then a and a' are associates.

Can anyone prove this?

Originally Posted by Samson

Hello all,

My book gave a theorem that stated that if a and a' both divide x and y, and that every g that divides x and y also divides a and a', then a and a' are associates.

Can anyone prove this?

a and a' are both gcd's of x and y. What does this tell you?

Does this mean that a = a'?

Originally Posted by Samson

Does this mean that a = a'?

Since a divides every other element that divides x and y, a must divide a'.

The same reasoning shows a' divides a.

Thus a and a' are associates by definition.

Thank you!