A simple proof?

Given: $a=b\times q+r,0\leq r < b$ .Yes,that is the division equation.Try to prove that, $GCD(a,b)=GCD(b,r)$ .Try not to use any theorem for solving this.Just use simple Algebra.This is from my class $10^{th}$ textbook,it was just written as a statement and no proof was given.I came up with one which I will post afterwards.

#NumberTheory #GreatestCommonDivisor(GCD/HCF/GCF)

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

To prove :- If $a=bq+r$ , then $(a,b)=(b,r)$ , where $(x,y)$ denotes GCD of x and y. Proof:- Let $(a,b)=d$ and $(b,r)=e$ . Then, (a,b) = $d \Rightarrow d|a$ and $d|b \Rightarrow d|a$ and $d|qb \Rightarrow d|(a-qb) \Rightarrow d|r$ . Hence, $d|r$ and $d|b \Rightarrow \boxed{d|(b,r)}$ . Similarly, (b,r) = $e \Rightarrow e|b$ and $e|r \Rightarrow e|bq$ and $e|r \Rightarrow e|(bq-r) \Rightarrow e|a$ . Hence, $e|a$ and $e|b \Rightarrow \boxed{e|(a,b)}$ . We see from the boxed expressions that $(a,b)|(b,r)$ and $(b,r)|(a,b) \Rightarrow (a,b) = (b,r)$ . Q.E.D.

Venkata Karthik Bandaru - 6 years, 2 months ago

Very nice proof!But how did you come up with it so fast?Have you seen similar problems before?

Adarsh Kumar - 6 years, 2 months ago

Yeah, I am pretty familiar with these kind of problems ! I came across problems involving proofs of gcd while doing number theory...

Venkata Karthik Bandaru - 6 years, 2 months ago

@Venkata Karthik Bandaru – So,is this your own?Do u have any NT books?

Adarsh Kumar - 6 years, 2 months ago

@Adarsh Kumar – Not my own bro, I just remember seeing this proof in some book ! Btw this is really important one, it is the basis of the Euclids Algorithm ! For good books, check out David A. Santos books, they are available in pdf format freely, and they are really good books to follow 🎓 📱 !

Venkata Karthik Bandaru - 6 years, 2 months ago

@Venkata Karthik Bandaru – Oh!Thanx for the recommendations!Thanx a lot!!Grateful!

Adarsh Kumar - 6 years, 2 months ago

@Venkata Karthik Bandaru – Yes,yes this is really an important one!It is the basis of Euclid's Algorithm,that is how I came across this.It must be in your book also.Check it!

Adarsh Kumar - 6 years, 2 months ago

I completed the proof in the same way - I personally remember the proof from "Elementary Number Theory " by Jones - It's a great book :)

Curtis Clement - 6 years, 2 months ago

Thanks a lot for your suggestion sir 👍 ! The book is really awesome for beginners 💎 !

Venkata Karthik Bandaru - 6 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$