Thanks for the hint! So, we've established that a|b, a|b, so a|b-c.
b-c = (4n+3) - (2n+1) = (2n+2)
Therefore, we know a|2n+1 and a|2n+2. These two are only one apart. Is that why a = +1 or -1?
So, I know K1=4n+3, where K is an integer, and I know K2=2n+1. But then, I have a, K1, and K2, and n, all of which are unkowns. How, then, do I solve for this?
Homework Statement
Suppose that n is an odd integer. Prove that n is either one greater than a multiple of 4 or one less than a multiple of 4.
Homework Equations
N/A
The Attempt at a Solution
I realize that this is going to be a direct proof. However, I am stumped on where to...
Homework Statement
For some integer n, a|(4n+3) and a|(2n+1). Therefore, 4n+3 is an integer multiple of a, as well as (2n+1). Prove or disprove that a=+/-1.
Homework Equations
N/A
The Attempt at a Solution
I have been working on this one for quite some time now, but I cannot...
Homework Statement
Every triple of consecutive odd natural numbers, with the first being at least 5, contains at least on composite.
Homework Equations
N/A
The Attempt at a Solution
I know from number theory that of every set of consecutive odd integers, one of them is divisible...