Q. 54.5( 2 Votes )

Prove that if 6 has no common factor with n, n2 — 1 is divisible by 6.

Answer :

Given here, there is no common factor between 6 and n.

Therefore, 6 and n are two distinct natural numbers.


We know that 6 has 2 and 3 as prime factors.


n can be written as n = 2k + 1 for all








Therefore, we can see that is divisible by 2.


Similarly, n can be written as n = 3k + 1 for all








Therefore, we can see that is divisible by 3.


Similarly, n can also be written as n = 3k–1 for all








Therefore, we can see that is divisible by 3.


So, n2 — 1 is divisible by 2 and 3 both.


Since, 2 and 3 are prime numbers.


Therefore, n2 — 1 is divisible by 2 × 3 = 6.


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