andrew.ng wrote:If {n} denote the remainder when 3n is divided by 2 then which of the following is equal to 1 for all positive integers n?
I/ {2n+1}
II/ {2n}+1
III/ 2{n+1}
A/ I only
B/ II only
C/ I and II
D/ III only
E/ II and III
I don't really understand the question. Could somebody help?
Thanks
As with most "wacky definition" questions, we need to start by really decoding the question stem.
Here, we're given this new operation {}. We're told that {n} equals the remainder when 3n is divided by 2. It's often helpful to do a couple of quick concrete examples to help you understand the operation.
So:
{2} = rem of (3*2/2) = rem of 6/2 = 0
{3} = rem of (3*3/2) = rem of 9/2 = 1
We can quickly determine that if n is odd, the remainder will be 1; if n is even, the remainder will be 0. Accordingly:
{odd} = 1
{even} = 0
Now to the exact question: which of the following is equal to 1 for all positive integers n?
We can quickly eliminate (III), since 2*(integer) is never going to equal 1 (even if we didn't understand the operation, we should be able to cross out III). III isn't part of the solution: eliminate (D) and (E).
The other two statements occur with equal frequency in A/B/C, so let's start with I, which looks a bit simpler:
I {2n+1}
Well, we know that 2n will be even, so 2n+1 will be odd. Based on our previous work, we know that {odd}=1, so (I) satisfies the question: eliminate B (only A and C left!).
II {2n} + 1
Again, we know that 2n will be even and that {even}=0. So, II is really:
0 + 1 = 1
and, consequently, II also satisfies the question.
I and II are both always equal to 1: choose C.
