Is any number coprime to itself?

Question

I have a fundamental question, we know that Coprime numbers are integers that have only 1 as their common factor. Now, my question is how it works for the same numbers, for example, is $10$ coprime to $10$? Since they can be simplified as $\frac{1}{1}$ and $1$ is coprime to $1$.

Answer 1

Your explanation for why $a$ should be coprime to $a$ is the exact reason that $a$ is not coprime to $a$ for $a > 1$.

In particular, positive integers $p,q$ are coprime if and only if the fraction $\frac pq$ cannot be simplified. The fact that $\frac {10}{10}$ can be simplified to $\frac 11$ means that the numbers $10,10$ are not coprime.

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On the other hand, every positive integer is coprime to $1$. Equivalently, the fractions $\frac 1a$ and $\frac a1$ cannot be simplified.

Answer 2

Two numbers $a$ and $b$ are coprimes to each other if $gcd(a,b) =1$. Since $gcd(a,a) = a$ we have $a$ is not coprime with itself for $a \ne 1$.