# Can a greatest common factor be 1?

Mar 28, 2016

Yes

#### Explanation:

For example, let $X = \left\{{p}_{1} , {p}_{2} , \ldots . , {p}_{n}\right\}$ be a set of prime numbers.

Then $\forall {p}_{i} \in X , 1 | {p}_{i} \mathmr{and} {p}_{i} | {p}_{i}$ .

That is, the only 2 factors of ${p}_{i}$ is $1$ and itself, $\forall i = 1 , 2 , \ldots . , n$.

Since ${p}_{i}$ is prime for all i,=>p_ does not divide into ${p}_{j} \forall i \ne j$.

Therefore, $H C F \left({p}_{1} , {p}_{2} , \ldots . , {p}_{n}\right) = 1$ .

Aug 26, 2016

It is easily possible to have only 1 as a common factor.

#### Explanation:

Numbers such as 24 and 35 do not have any common factor apart from 1.

Their prime factors are as shown:

$24 = 2 \times 2 \times 2 \times 3$
$35 = \textcolor{w h i t e}{\times \times \times \times \times} 5 \times 7$

In this case the HCF would be 1.

1 is always a factor of every number.