How do I prove that 15462227 and 15462229 relatively prime?

Question

Are 15,462,227 and 15,462,229 relatively prime?

How do I construct a proof for this.

I know that for n and n+2 to be relatively prime it is the case that n has to be odd. Is this the only requirement for these numbers to be relatively prime?

Answer 1

Hint: $\gcd(a,b)=\gcd(a,b-a)$.

Answer 2

Calculate the GCD by Euclid's algorithm! The GCD of n and n+2 is the GCD of 2 and n, which is 1 if n is odd.

Answer 3

In general the way to answer this question is the Euclidean algorithm. In this case it's very simple (one step):

$$ 15,462,229 = 15,462,227 + 2 $$

so the greatest common divisor of the two numbers in question is the same as the greatest common divisor of either and $2$. That's clearly $1$.

Answer 4

If $15,462,229$ and $15,462,227$ have a common divisor $d$ they can be written as $q_1d$ and $q_2 d$, and so is their difference $(q_1-q_2)d = 2$, so $d$ is also a divisor of $2$ thus either $2$ or $1$, which one is it?