What Are the Key Divisibility Rules and Counterexamples in Number Theory?

[1+1=1]

hey i have more problems that can really exercise the mind! here are 3.

1. prove if q is divisible by (r +s) then either q is divisible by r or q is divisible by s.

2. if d>0, (fd+ed) = d(f,e). proof.

3. a divisible by b => a^m divisible by b^m a,b,m are in Z+.

i think i have some thoughts and i will share.

1. say that (b+c) = ma, where m is in Z+. after that, i am guessing...

2. could use contradiction here. say that d=0, then show the proof? anyone has any takes on this?

3. there exist m and n s.t. bn=ma. lost after here.

anyone w/ information/thoughts please share.

 

[Muzza]

For number 3, just apply the definition of "divisibility"... There is a k such that a = bk, and thus a^m = (bk)^m = b^m * k^m, and so a^m / b^m = k^m \in Z, hence b^m | a^m.

 

[matt grime]

what does the notation in 1 mean, are you talking about ideals?

the second follows, i believe, if you show the RHS divides the LHS and the LHS divides the RHS

 

[Gokul43201]

What does (a+b) denote ?

 

[AlMacD]

Number 1: By (r+s), do you mean the sum of r and s? If so, a quick counterexample: 25 is divisible by (2+3) but it's not divisible by 2 or by 3.

 

[Zurtex]

Number 1: By (r+s), do you mean the sum of r and s? If so, a quick counterexample: 25 is divisible by (2+3) but it's not divisible by 2 or by 3.

Bah, I was just about to put the exact same counter example up