Ian Stewart wrote:

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

1)n is odd
2)n is not divisible by 8

You might notice that this is a difference of squares:

n^2 - 1 = (n+1)(n-1)

If n is odd, then n-1 and n+1 are consecutive even integers. If you take any two consecutive even integers, one of them will be divisible by 4, the other not, so their product must be divisible by 8. Thus, if we know n is odd, we can be certain that n^2 -1 will be divisible by 8, and r will be zero. 1) is sufficient. 2) is not; n might be even, or might be odd. A.

I knew it was surely answer A because I tried with some odđ numbers but this is full explanation for this.

If you cannot think of any direct method,
you can always substitute numbers and check each of the staetments

First statement says "n is odd"
Let's substitute few odd numbers
eg: 3 => ((3^2) -1)/8 gives a remainder of 0
7 => ((7^2) -1)/8 gives a remainder of 0
11 => ((11^2) -1)/8 gives a remainder of 0
Basically if we pick any odd number here we always get the remainder(r) as 0.
So statement (a) is sufficient.

Now let's check statement (b) which says "n is not divisible by 8"
Lets substitute both odd and even numbers.
eg: 10 => ((10^2) -1)/8 gives a remainder of 3
12 => ((10^2) -1)/8 gives a remainder of 3
Re-writing the odd numbers which we already checked
11 => ((11^2) -1)/8 gives a remainder of 0
7 => ((7^2) -1)/8 gives a remainder of 0
So remainder 'r' can be either of '0' or '3' here depending on whether 'n' is odd or even.
Hence statement (b) is not sufficient.

mana6 wrote:If n is a positive integer and r is the remainder when (n^2)-1 divided by 8 ,what is the value of r?


I do not understand how the answer is a. i did this to get e as my answer:

1) n is not odd

n q r
4 1 7
6 4 3
8 7 7

This statement is insufficient because it doesnt definitively give you one answer for r. or so i thought

2) n is not divisible by 8

n q r
12 17 7
10 12 3
7 6 0

this again doesnt definitively give you one value for r so i said this was also insufficient.

1and 2) n is not odd and it is not divisible by 8

n q r n q r
2 0 3 6 4 3
4 1 7 10 12 3

when i tried twelve it also gave me a remainder of seven so since r can be three or seven this is also not enough. is there something im missing?

n is odd. That might make a difference. If not, let me know and I'll try to help.

If you don't know how to spot the difference of 2 squares, here's another method:

1) n is odd (therefore it can't be divisible by 8 anyway, so the second statement is redundant)

Let p be any positive integer

n = 2p - 1 is the definition of an odd number

So n^2 - 1 = (2p - 1)^2 - 1 = 4p^2 -4p + 1 - 1 = 4p(p - 1)

Consider 2 cases:

(A) p is odd: 4p(p-1) = 4p x even = 8 x something, so dividing by 8 gives remainder 0
(B) p is even: 4 x even (p - 1) = 8 x something, so dividing by 8 gives remainder 0

Therefore the remainder is 0 for all situations.

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

1)n is odd
2)n is not divisible by 8

asnwer a st 1 is enough to answer the question while st 2 is not sufficient

The answer is A
If n is odd, r is always zero
In the other case, try n = 6 or n = 12, the value of r is diff in each case

The answer should be (C)
If n is divisible by 8 the remainder will not be 0.
In all other cases it would be 0.

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

1)n is odd
2)n is not divisible by 8

Statement 1)

if n=1, n^2-1=0 and r=0
if n=3, n^2-1=8 and r=0
if n=5, n^2-1=24 and r=0
if n=7, n^2-1=48 and r=0
Consistent value of 'r' therefore SUFFICIENT

Statement 2) n is not divisible by 8

if n=1, n^2-1=0 and r=0
if n=2, n^2-1=3 and r=3

Inconsistent value of 'r' therefore INSUFFICIENT

Answer Option: A

a is the option

Answer E

Answer : A

Answer. Statement 1 is sufficient.

Answer. Statement 1 is sufficient.

A

1. n odd --> n=2k+1 --> n^2-1=(n-1)*(n+1)=(2k+1-1)*(2k+1+1)
= 4k(k+1)
k(k+1) : 2 (remainder=0) --> 4k(k+1):8 --> sufficient
2. n is not divisible by 8
Ex: n=1 --> n^2-1=(n-1)(n+1)=0 --> divisible by 8
n=2 --> n^2-1 is not divisible by 8 --> insufficient


--> A

A