When do I know a fraction is in lowest terms?

Question

How can I tell when a fraction is in lowest terms. Example;

$\frac{6599}{9999}$

Answer 1

A fraction $\frac{a}{b}$ is in lowest terms when $a$ and $b$ have no common divisor but $1$ (and $-1$). That is $a$ and $b$ are coprime or said differently their greatest common divisor is $1$.

For small numbers an efficient can be to decompose the two numbers $a$ and $b$ into a product of primes and to see if there are common factors. For larger numbers other ways to determine the GCD are more efficient; in particular you could uses Euclid's alogorithm.

Answer 2

$$\gcd(6599,9999)=\gcd(3400,9999)=\gcd\left(2^3\cdot 5^2\cdot 17,3^2\cdot 11\cdot 101\right)=1.$$

Answer 3

HINT: A fraction $\frac{p}{q}\ \ \forall \ \ p, q\in I, \ q\neq 0$ is said to be in the lowest terms when $p$ & $q$ are co-prime integers i.e. having no common factor ($>1$).

Hence, for given fraction $\frac{6599}{9999}$, factorize both the numerator $6599$ & the denominator $9999$ to check if there is any common factor

Answer 4

Each common divisor is a factor of the difference $9999-6599=3400$. Since $3400$ prime factors are $2$, $17$ or $5$ there's no chance to cancel.

Answer 5

When the numerator and denominator are the smallest possible integers (or if there are negative numbers involved, the absolute values of the numerator and denominator) while keeping the same value. For example: $$\frac{256}{144} = \frac{128}{72} = \frac{64}{36} = \frac{32}{18} = \frac{16}{9}.$$ These fractions all mean the same thing, but only the last one is in lowest terms. Notice on the first one that 256 and 144 share several common divisors, the greatest of which is 16. So we can divide them both by 16 to get 16 and 9 as numerator and denominator, and at this point, since $\gcd(16, 9) = 1$, the fraction can't be put in lower terms than that.

But with $\frac{6599}{9999}$, the fraction is already in lowest terms. Among their positive divisors, 6599 and 9999 share only one divisor: 1. The fraction can't be simplified further.