If p is a positive integer, what is the remainder when p^2 – 1

If p is a positive integer, what is the remainder when \(p^2 – 1\) is divided by 8?

(1) p is completely divisible by 3
(2) p leaves a remainder when divided by 2.

Question stem analysis :-

is (p+1)(p-1) is divisible by 8?

if p = odd => (Odd+1)(Odd-1) = Even*Even = Even

Minimum value of p as odd = 1 => (p+1)(p-1) = 0 => divisible by 8
Next values of p as odd = 3 => (3+1)(3-1) = 4*2 => divisible by 8

if p = even => (Even+1)(Even-1) = Odd*Odd = Odd => Never divisible by 8

Question :- is p odd or even?

i) p = 3m => p = 3 => \((p^2-1)\) divisible by 8; p=12 => \((p^2-1)\) - Not divisible by 8 - Insufficient

ii) p leaves a remainder when divided by 2 => Remainder = 1 (As remainder can be only 0 or 1) => p is odd - Sufficient per the question stem analysis

Answer - B

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Excellent video. Thanks a lot for your help

If we have p as 1, then based on second statement, the p^2-1 will become 0 and leave a remainder of 8. Help?