Prove if x*y = 1 then x = y = 1

Assume that x and y are positive integers, prove that if x * y = 1 then x = y = 1

If x * y = 1 then x = (1/y) and y = (1/x)

Yesterday we proved that if an integer x divides an integer y then x<=y.

So since y divides 1 or (1/y) then y<=1

and since x divides 1 or (1/x) then x<=1

Since both x and y are integers then y=1 and x=1, so x = y = 1

Prove that if x divides y, then x<=y

Assume that x and y are positive integers, prove that if x divides y then x<=y.

If x divides y then there exists another integer j such that y=xj.


Now we can say x =< xj and canceling out the x we get 1<=j or that is to say, j is greater than or equal to 1.

This is true because both x and y are positive so j must be positive for our initial antecedent y=xj to be true. Also since we assumed j is an integer then j>=1. So now we know that x<=xj and by substitution of y=xj we get x<=y.