Euclid's theorem states that for any two positive integers m and n, the greatest common divisor (GCD) of m and n is equal to the GCD of n and the remainder of m divided by n. In this question, we will verify Euclid's theorem for the given function u(x,y)=(x^n/2y^n/2)(x^ky^k).


To verify Euclid's theorem for the given function, we need to show that the GCD of x^n/2y^n/2 and x^ky^k is equal to the GCD of x^ky^k and the remainder of x^n/2y^n/2 divided by x^ky^k.

Let d be the GCD of x^n/2y^n/2 and x^ky^k. Then, we can write:

x^n/2y^n/2 = ad
x^ky^k = bd

where a and b are coprime integers.

Since a and b are coprime, we can write:

x^n/2y^n/2 = a(x^ky^k)(x^(n-k)/2y^(n-k)/2)

Since a and b are coprime, a must divide x^(n-k)/2y^(n-k)/2. Therefore, a is a common divisor of x^ky^k and x^n/2y^n/2. Hence, a divides d.

Similarly, we can write:

x^n/2y^n/2 = b(x^ky^k)(x^(n-k)/2y^(n-k)/2) + r

where r is the remainder of x^n/2y^n/2 divided by x^ky^k. Since a and b are coprime, b must divide r. Therefore, b is a common divisor of x^ky^k and r. Hence, b divides d.

Therefore, d is the GCD of x^ky^k and the remainder of x^n/2y^n/2 divided by