Proving the Integer Multiples of A for (4n+3) and (2n+1): Homework Statement

[nelson98]

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 even start it right. Can somebody help me?

 

[tiny-tim]

hi nelson98!

if a|(4n+3) and a|(2n+1), then what else does a divide?

 

[micromass]

To enhance on tiny-tim's hint. Does a divide certain sums/differences??

 

[nelson98]

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?

 

[micromass]

Hint: if a|b and a|c, then a|b-c. Apply this several times...

 

[nelson98]

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?

 

[micromass]

Yeah.

Or you could just do the same thing again to b=2n+2 and c=2n+1. That would also give you the answer...

 

[nelson98]

Thank you very much for the help! I appreciate you working me through this.