If p is a positive integer and r is the remainder when p2 1 is divide

Question

If p is a positive integer and r is the remainder when p^2-1 is divided by 8, what is the value of r ?

(1) When p is divided by 4, the remainder is zero.
(2) When p is divided by 8, the remainder is not zero.

danlew · Answer

Plugging in some numbers for P to check conditions.

A--> P=4, 8, 12 etc all divisible by 4, but (P^2-1) gives reminder as 7 always. SUFFICIENT
B--> If P=2, (P^2-1) reminder is 3. But, if P=3, (P^2-1) reminder is 0. Two different answers. INSUFFICIENT

Answer --> A

laddaboy · Answer

Plugged in numbers to solve the question

Given p>0 and P^2-1/8 = quotient + r

From

1) Using p=4,8,12 etc we see the reminder is constant(7). Sufficient

2) Here p can be 1,2,3,4,5,6,7 but for each value of - the reminder will change. Hence No sufficient

Hence A

niks18 · Answer

p^2-1 = 8q+r , or  p^2 = 8q+r+1

Statement 1 : implies that  p=4k , so  p^2 = 16k^2 
This means  p^2  is divisible by  8 . Hence  r+1=8 , or  r=7 .  Sufficient

Statement 2 : here the remainder can be anything from 1 to 7. Hence  Insufficient

Option  A

kunalsinghNS · Answer

Hey ! I was able to solve this question by taking values but can you tell how did you get 
r+1 = 8 ?  niks18

niks18 · Answer

Hi  kunalsinghNS

We know that  p^2  is divisible by  8  and

p^2 = 8q + (r+1) . As  8q  is divisible by  8 , so  (r+1)  has to be divisible by  8  to make  p^2  divisible by  8

here  r  is a remainder when  p^2-1  was divided by  8  hence  r<8 , so the maximum value  (r+1)  can take is  8

Hence  r+1=8 , or  r=7

kunalsinghNS · Answer

Thank you  niks18

hellosanthosh2k2 · Answer

(p^2 - 1)  % 8 = ?

Statement 1:
 Let p = 4m (since it is divisible by 4)
  (4m - 1)(4m + 1 ) % 8 = ?
 
  16m^2 - 1  => since  16m^2  is multiple of 8, any value 1 less than multiple of 8, when divided by 8 will yield remainder 7.

Sufficient

Statement 2:
 p is even but not divisible by 8,  (p-1)(p+1 ) => (odd * odd)  %8 => can lead to any value between 0 and 7
 p is odd , say p = 9, then  (p-1)(p+1)  => (8 * 10) % 8 => remainder 0

InSuff

Answer (A)

If p is a positive integer and r is the remainder when p2 1 is divide

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