It would greatly appreciate help with the following problem!
If n is a positive integer and r is the remainder when n^2 - 1 is divisible by 8, what is the value of r?
1) n is odd
2) n is not divisible by 8
The correct answer is Statement 1 alone is sufficient, but statement 2 alone is not sufficient.
Could you please also explain whether there are any special rules about the remainder?
(all I know is: y = ax +b, the same as y/a = x +b/a, where x and b/a are integers).
May be there are some great rules which could safe time and help to succeed on the exam? I could not find explanation about remainder in "Number Properties" book and I would be happy to get some help with that. Thanks!
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- RonPurewal
- Students
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- Joined: Tue Aug 14, 2007 8:23 am
If you've got something this obscure, you might just want to try a bunch of cases and see if a pattern emerges. (Pattern recognition is much more powerful than memorization of any # of rules!)
To wit:
N ..... N^2 - 1 ..... Remainder
1 ..... 0 ..... 0
2 ..... 3 ..... 3
3 ..... 8 ..... 0
4 ..... 15 ..... 7
5 ..... 24 ..... 0
6 ..... 35 ..... 3
7 ..... 48 ..... 0
8 ..... 63 ..... 7
There's your pattern: 0, 3, 0, 7, 0, 3, 0, 7, etc.
(1) Notice that all the odds give a remainder of 0, so you know R is 0. Sufficient.
(2) None of 1 through 7 are divisible by 8, and, as you've seen, those yield 3 different values of R. Not sufficient.
There ARE ways in which you can figure these things out theoretically. For instance:
If N is odd, then N^2 - 1 = (N + 1)(N - 1). That's a product of two even numbers, one of which is a multiple of 4 (because every other even # is a multiple of 4), so the overall thing is a multiple of 8. Thus remainder = 0 all the time.
But why would you want to do that, when the pattern jumps out at you after a few plug-ins?
don't be afraid to experiment (as in throwing in numbers and watching for patterns).
To wit:
N ..... N^2 - 1 ..... Remainder
1 ..... 0 ..... 0
2 ..... 3 ..... 3
3 ..... 8 ..... 0
4 ..... 15 ..... 7
5 ..... 24 ..... 0
6 ..... 35 ..... 3
7 ..... 48 ..... 0
8 ..... 63 ..... 7
There's your pattern: 0, 3, 0, 7, 0, 3, 0, 7, etc.
(1) Notice that all the odds give a remainder of 0, so you know R is 0. Sufficient.
(2) None of 1 through 7 are divisible by 8, and, as you've seen, those yield 3 different values of R. Not sufficient.
There ARE ways in which you can figure these things out theoretically. For instance:
If N is odd, then N^2 - 1 = (N + 1)(N - 1). That's a product of two even numbers, one of which is a multiple of 4 (because every other even # is a multiple of 4), so the overall thing is a multiple of 8. Thus remainder = 0 all the time.
But why would you want to do that, when the pattern jumps out at you after a few plug-ins?
don't be afraid to experiment (as in throwing in numbers and watching for patterns).
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