If #a# and #b# are coprime and #c# is a factor of #a#, then prove that #b# and #c# too are coprime?

Question

Aurora Arias · Accepted Answer

Please see below.

Numbers whose GCD is #1# are known as coprime or relatively prime. If two numbers are prime numbers, they will be coprime. But even if there are two composite numbers, let us say #25# and #36#, as there is no common factor between them, they are coprime and their GCD is #1#. Now as GCD of #a# and #b# is #1#, there is no common factor between them. We have #c#, which is a factor of #a#, but as there is no common factor between #a# and #b#, there will not be a common factor between #b# and #c#, as otherwise this common factor would have been GCD of #a# and #b#, (as it divides both). Hence, GCD of #b# and #c# too is #1#. As an example, GCD of #36# and #25# is #1#, but though #12# is a factor of #36#, GCD of #12# and #25# too is #1#

Roman Alvarez · Answer

undefined To prove that $b$ and $c$ are coprime, we'll use the definition of coprime numbers, which states that two numbers are coprime if their greatest common divisor (GCD) is 1.

Given that $a$ and $b$ are coprime, and $c$ is a factor of $a$, we can express $a$ as $a = c \times k$ for some integer $k$.

Now, if $b$ and $c$ have a common divisor other than 1, then that divisor must also be a divisor of $a$, because $a = c \times k$. However, since $a$ and $b$ are coprime, they do not share any common factors other than 1. Therefore, $b$ and $c$ must also be coprime.