Show any natural number greater than 3 that may be written a

Show any natural number greater than 3 that may be written as n^2 - 1 for some n must be composite. A number of the form n^2 - 2 can never be divisible by 4.

Solution

b) Given n²-1

Need to show that any natural number greaterthan 3 which can be written as n²-1 for some n must be a composite.

Composite: A number which divide by any other number along with 1 and itself.

n²-1 :

Here, n>2 to satisfy the given statement.

When \'n\' is a even number,

n² is also a even number, n²-1 is an odd number and it will be divide by either 2 or 5.

When \'n\' is an odd number,

n² is also a odd number, n²-1 is an even number and all even numbers can be divide by 2.

For any number \'n\', n²-1 is divide by either 2 or 5.(n>2 for the given statement).so, n is composite.

Therefore, for any natural number greaterthan 3 which can be written as n²-1 for some n must be a composite.

c) Need to show that n²-2 can\'t be divisible by 4.

When \'n\' is a even number,

n², square of any even number is divisible by 4.

As n² is divisible by 4, n²-2 can\'t be divisible by 4 because it results in a remainder 2.

When \'n\' is an odd number,

n² is also a odd number.

n²-2 is an odd number and odd numbers can never be divisible by 4.

For any number \'n\', n²-2 can never be divisible by 4