Problem 12E

Give a reason for your answer. Assume that all variables represent integers.

Exercise

If n = 4k + 1, does 8 divide n2 − 1?

Solution:-

Step1

Given that

We have to give a reason for our answer. Let all variables represent integers. We have to If n = 4k + 1, does 8 divide n2 − 1?

Step2

8 divide − 1 :-yes

Using definition of divisibility, if m and e are integers, then m is divisible by e only if m=ec, for some integer c.

Suppose that n=4k+1

Put value of n in − 1 we get

− 1=

=

=

=

Step3

From the definition of divisibility

And is an integer because k is an integer and sums and product of integers are integers.

Therefore, − 1 is divisible by 8.