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

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

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Answer 1 · 2017-03-18T04:56:18

Please see below.

Explanation:

Numbers whose GCD is $1$ $1$ 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 $25$ $25$ and $36$ $36$ , as there is no common factor between them, they are coprime and their GCD is $1$ $1$ .

Now as GCD of $a$ $a$ and $b$ $b$ is $1$ $1$ , there is no common factor between them.

We have $c$ $c$ , which is a factor of $a$ $a$ , but as there is no common factor between $a$ $a$ and $b$ $b$ ,

there will not be a common factor between $b$ $b$ and $c$ $c$ , as otherwise this common factor would have been GCD of $a$ $a$ and $b$ $b$ , (as it divides both).

Hence, GCD of $b$ $b$ and $c$ $c$ too is $1$ $1$ .

As an example, GCD of $36$ $36$ and $25$ $25$ is $1$ $1$ , but though $12$ $12$ is a factor of $36$ $36$ , GCD of $12$ $12$ and $25$ $25$ too is $1$ $1$

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

1 Answer

Explanation:

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If aa and bb are coprime and cc is a factor of aa, then prove that bb and cc too are coprime?

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