MHB GCD Discrete Math: Proving GCD(a,b)=1

ssome help
Messages
3
Reaction score
0
Given that GCD(na,nb) = n * GCD(a,b) for a,b,n ∈ Z+

a) Prove that, if GCD(a,b) = 1 then GCD(a+b, a-b) = 1 or GCD(a+b,a-b) = 2
Hint: Let D = GCD(a+b, a-b), show that D | 2a and D | 2b thus D | GCD(2a,2b) then use the given
b) Prove that, if GCD(a,B) = 1, then GCD(2a+b, a+2b) = 1 or GCD(2a+b, a+2b) = 3

Any assistance would be great.
 
Physics news on Phys.org
ssome help said:
Given that GCD(na,nb) = n * GCD(a,b) for a,b,n ∈ Z+

a) Prove that, if GCD(a,b) = 1 then GCD(a+b, a-b) = 1 or GCD(a+b,a-b) = 2
Hint: Let D = GCD(a+b, a-b), show that D | 2a and D | 2b thus D | GCD(2a,2b) then use the given
b) Prove that, if GCD(a,B) = 1, then GCD(2a+b, a+2b) = 1 or GCD(2a+b, a+2b) = 3

Any assistance would be great.

Welcome to MHB, ssome help!

Nice problem. ;)

Did you try anything?
When you show something you tried, or if you explain where you are stuck, we can help you to solve and understand this.
 
The key fact to use here is that if d | x and d | y, then d divides any linear combination of x and y, i.e., d | (mx + ny) for any integer m and n.
 
Setting...

$\displaystyle x = a + b$

$\displaystyle y = a - b$ (1)

... solving (1) we obtain...

$\displaystyle a = \frac{x + y}{2}$

$\displaystyle b = \frac{x - y}{2}$ (2)

Now if x and y have a common factor different than 2, then x + y and x - y have the the same common factor and the same would be for a and b and that is a contradiction...

Kind regards

$\chi$ $\sigma$
 
Hi all, I've been a roulette player for more than 10 years (although I took time off here and there) and it's only now that I'm trying to understand the physics of the game. Basically my strategy in roulette is to divide the wheel roughly into two halves (let's call them A and B). My theory is that in roulette there will invariably be variance. In other words, if A comes up 5 times in a row, B will be due to come up soon. However I have been proven wrong many times, and I have seen some...
Thread 'Detail of Diagonalization Lemma'
The following is more or less taken from page 6 of C. Smorynski's "Self-Reference and Modal Logic". (Springer, 1985) (I couldn't get raised brackets to indicate codification (Gödel numbering), so I use a box. The overline is assigning a name. The detail I would like clarification on is in the second step in the last line, where we have an m-overlined, and we substitute the expression for m. Are we saying that the name of a coded term is the same as the coded term? Thanks in advance.
Back
Top