Question

3) Is it possible for a prime p to divide both n and n+1? (for n > 0 and integer).

Answer 1

Answer is:
NO

Statement: It is not possible for a prime number p to divide both n and n+1

Proof:

Given p is a prime number.

Case 1:

p divides n                      (1)

To prove: p does not divide n+1

Proof:

Because of (1), n is a multiplication of p.

i.e.,

n = pk                  (2)

Because of (2), p cannot pk + 1, since pk + 1 is not a multiple of p.

Case 2:

p divides n+1.

Then,

n+1 = pk.

So,

n = pk -1

So, p cannot divide n, because pk - 1 is not a multiple of p.

Combining both Case 1 & 2:

We prove the theorem that if p is prime, then p cannot divide both n and n+1