Use Theorem and the result of exercise 10 to prove that if p is any prime number with p ≥...

Use Theorem and the result of exercise 10 to prove that if p is any prime number with p ≥ 5, then the sum of squares of any p consecutive integers is divisible by p.

Theorem 1

Sum of the First n Integers

For all integers n ≥ 1,

Proof (by mathematical induction):

Let the property P(n) be the equation

Show that P(1) is true:

To establish P(1), we must show that

But the left-hand side of this equation is 1 and the right-hand side is

also. Hence P(1) is true.

Show that for all integers k≥ 1, if P(k) is true then P(k + 1) is also true:

[Suppose that P(k) is true for a particular but arbitrarily chosen integer k ≥ 1.

That is:] Suppose that k is any integer with k ≥ 1 such that

[We must show that P(k + 1) is true. That is:] We must show that

or, equivalently, that

[We will show that the left-hand side and the right-hand side of P(k + 1) are equal to the same quantity and thus are equal to each other.]

The left-hand side of P(k + 1) is

And the right-hand side of P(k + 1) is

Thus the two sides of P(k + 1) are equal to the same quantity and so they are equal to each other. Therefore the equation P(k + 1) is true [as was to be shown].

[Since we have proved both the basis step and the inductive step, we conclude that the

theorem is true.]