Question

Докажите ,что (x^n-a^n)⋮(x^k-a^k), если n⋮k, n∈N , k∈N

Answer 1

Кстати, верен и обратный результат.
Воспользуемся формулой
x^m-y^m=(x-y)(x^(m-1)+x^(m-2)y+ x^(m-3)y^2+...+y^(m-1))

По условию n=kt. Поэтому
x^n-a^n=x^(kt)-a^(kt)=(x^k)^t-(a^k)^t= (x^k-a^k)((x^k)^(t-1)+... (a^k)^(t-1))
Отсюда и следует требуемый результат 

Answer 2

Proof that (x^n-a^n)⋮(x^k-a^k) if n⋮k, n∈N, k∈N

To prove that (x^n-a^n)⋮(x^k-a^k) when n⋮k, where n and k are natural numbers, we can use the concept of divisibility.

Let's assume that n⋮k, which means that n is divisible by k. This implies that there exists an integer m such that n = mk.

Now, let's consider the expression (x^n-a^n). Using the identity a^n - b^n = (a-b)(a^(n-1) + a^(n-2)b + a^(n-3)b^2 + ... + b^(n-1)), we can rewrite the expression as:

(x^n-a^n) = (x^(mk)-a^(mk)) = (x^k)^m - (a^k)^m

Now, let's define