GCD property

Questions asks to confirm the property.

#NumberTheory #GreatestCommonDivisor(GCD/HCF/GCF) #Proofs #Divisibility

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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

$c | (a+b) \rightarrow c|a \wedge c|b$ is wrong. Easily counterexample as $a = 4, b = 5, c = 3)$ .

Samuraiwarm Tsunayoshi - 6 years, 6 months ago

Hint: use Bezout's identity

Samuraiwarm Tsunayoshi - 6 years, 6 months ago

Another hint: $(a,b) = 1$ if and only if $ax+by = 1$ .

Samuraiwarm Tsunayoshi - 6 years, 6 months ago

@Samuraiwarm Tsunayoshi – Hey,@Samuraiwarm Tsunayoshi check my sol in Kartik Sharma's solution's replay box.

shivamani patil - 6 years, 6 months ago

Let $a = {{p}_{1}}^{{e}_{1}}{{p}_{2}}^{{e}_{2}}{{p}_{3}}^{{e}_{3}}..... {{p}_{n}}^{{e}_{n}}$

Now, $a = 0 mod {p}_{i}$

We know that gcd(a,b) = 1, so none of the ${p}_{i}$ divide b

$cn = a + b$

$cn mod {p}_{1} = 0 + x$

$cn = x mod {p}_{1}$

And so on, till ${p}_{n}$ which tells us that ${p}_{i}$ doesn't divide c. Therefore there is no common divisor of c and a, so, $gcd(a,c) = 1$

Similarly we can show for b.

@shivamani patil Check out and now I think it is fine.

Kartik Sharma - 6 years, 6 months ago

If gcd(a,b) then $ax+by=1$ also $cn=a+b$ for some $n$ because $c|a+b$ .Therefore $cn-b=a$ ,substituting $cn-b=a$ in first equation we get $x(cn-b)+by=1=cnx-bx+by=c(nx)+b(y-x)$ which implies gcd(c,b)=1.Similarly gcd(c,a)=1.@Kartik Sharma how is it??Your's is good.

shivamani patil - 6 years, 6 months ago

Yep, looks good.

Samuraiwarm Tsunayoshi - 6 years, 6 months ago

I knew about this method too but that is already discussed by Samuraiwarm Tsunayoshi.

Kartik Sharma - 6 years, 6 months ago

@Kartik Sharma – But I solved it independently @Kartik Sharma

shivamani patil - 6 years, 6 months ago

Note that $a|b$ implies that there exists an integer $k$ suck that $ka = b$ , not $kb = a$ . So your assumption that $c = n(a+b)$ is wrong. It should be $cn = a+b$ .

Siddhartha Srivastava - 6 years, 6 months ago

Oh damn yes ! I knew it but just a silly mistake. Now, I am gonna edit it.

Kartik Sharma - 6 years, 6 months ago

@Kartik Sharma – Looking forward to your solution!

Samuraiwarm Tsunayoshi - 6 years, 6 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}`	$\boxed{123}$