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Question Number 216875 by ArshadS last updated on 23/Feb/25

Let  p  be a prime number greater than 3. Prove that  p^2 − 1    is  always  divisible by 24.

Letpbeaprimenumbergreaterthan3.Provethatp21isalwaysdivisibleby24.

Answered by maths2 last updated on 23/Feb/25

(p−1)(p+1)  3∣(p−1)(p+1);since p≡1,2[3]  p≡1,3,5,7[8]  ⇒p^2 ≡1[8]⇒8∣p^2 −1  since 3 and 8 are coprim 3.8=24∣p^2 −1

(p1)(p+1)3(p1)(p+1);sincep1,2[3]p1,3,5,7[8]p21[8]8p21since3and8arecoprim3.8=24p21

Commented by ArshadS last updated on 23/Feb/25

Nice! Thanks sir!

Nice!Thankssir!

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