Question about Rational numbers and prime numbers

Question

Suppose $r$ is a rational number; then we can express $r$ as $r = \dfrac pq,$ where $p,q$ are integers and $q>0$, and also $p$ and $q$ are relatively prime.

What does "relatively prime" mean?

Answer 1

$p$ and $q$ are relatively prime (or coprime) means $p$ and $q$ share no common factors (except $1$).

For example, $16=2\times2\times2\times2$ and $21=3\times7$ are relatively prime, so $\dfrac{16}{21}$ cannot be simplified.

On the other hand, $16=2\times2\times2\times2$ and $18=2\times3\times3$ share the factor $2$, so $\dfrac{16}{18}=\dfrac89$.

Answer 2

Integers $a$ and $b$ are called relatively prime, or coprime, if $\operatorname{gcd}(a, b)=1$. The notation $p,q$ usually denotes prime numbers (but not always), so I would write $$ r=\frac{a}{b}=\frac{\frac{a}{d}}{\frac{b}{d}}=\frac{a'}{b'}, $$ with $d=gcd(a, b)$ and $gcd(a',b')=1$. For example,

$$ \frac{143}{1243}=\frac{11\cdot 13}{11\cdot 113}=\frac{13}{113}. $$

Answer 3

Two integers are relatively prime if their greatest common divisor is $1$. This implies that $r$ is presented as a fraction $\frac pq$ where numerator and denominator cannot be further reduced through division by a common factor (as the only ones would be $1$ and $-1$).