Question

Prove that if 2|(a-b) and 3|(b-a), then 6|(a-b)

Answer 1

\(2 | (a-b) \implies a-b = 2k\)

\(3 | (b-a) \implies b-a = 3m\)

Answer 2

since \(\gcd(2, 3) = 1\) we can write \(2x + 3y = 1\) for some integers \(x\) and \(y\)

Answer 3

\(a-b = (a-b)\times 1 = (a-b)(2x+3y) = (a-b)2x + (a-b)3y\)

\(= -3m*2x + 2k * 3y = 6(-mx+ky )\)

so, \(6 | a-b\)

Answer 4

let me knw if smthng doesnt make sense

Answer 5

Thank you very much. Why does gcd(2,3)=1 imply that 2x + 3y =1 ?

Answer 6

good question :)

gcd of any two integers is the LOWEST possible positive integer that can be written as linear combination of those two integers

Answer 7

thats mouthful definition

Answer 8

you need to prove it upfront, before pursuing this current problem

Answer 9

gcd(a, b) = smallest possible positive integer than can be represented as "ax + by"

Answer 10

since gcd(2, 3) = 1
we can find x and y such that 2x + 3y = 1

Answer 11

That's very helpful, thank you!

Answer 12

np.. u wlc :)

in general :

if \(a|c\) ,\(b|c\), and \(\gcd(a, b) = 1\), then \(ab | c\)

Answer 13

I miss number theory, this stuff is so fun.:)

Answer 14

yes :) and after seeing the kind of problems u post... i think NT is relatively easy compared to abstract algebra or analysis lol

Answer 15

I love the abstract, and despise the analysis. Don't tell eliassaab :)

Answer 16

well actually I dont mind it if its new, but when we look at the same thing through 20 different ways I get .....

Answer 17

I'm sure NT gets nutty at some point, I have only taken a 300 level intro class.

Answer 18

ahh i never tried abstract/analysis lol @eliassaab

elementary NT is fun and easy to understand... advanced nt is harder than abstract and everything i feel... i had a look once and realized soon enough that it was not for meh

Answer 19

You can also argue this way.
2 occurs in the prime factorization of ( a-b)
3 occurs in the prime factorization of ( a-b) then\[
(a-b)=2^n 3^n q \, \text { where } n\ge 1; m\ge 1

\]
Hence 6 divides (a-b)

Answer 20

looks much simpler ! thnks @eliassaab :)

Answer 21

ps @eliassaab I got a 93% on the final and an A in the class

thanks for all the help this term in measure theory:)

Answer 22

YW. Glad you got an A @zzr0ck3r