# Question #585c4

Jul 11, 2017

$1$

The only way that the GCF of two numbers could be their product is if both numbers are 1 (that way the product isn't bigger than either of the numbers). And the LCM of 1 and 1 is $1$.

#### Explanation:

Hmm... factors of a number cannot be bigger than the number itself. In mathematical terms,

$\text{number " >= " factor of that number}$

But, we see here that we have two positive integers $a$ and $b$, such that:

$a b$ is a factor of both $a$ and $b$

This means that $a \ge a b$ and $b \ge a b$

We can do a bit of algebra to determine what these two numbers must be:

$a \ge a b \text{ "" "and" "" } b \ge a b$

Divide both sides of the first inequality by $a$, and the second by $b$:

$\frac{a}{a} \ge \frac{a b}{a} \text{ "" " and " "" } \frac{b}{b} \ge \frac{a b}{b}$

$1 \ge b \text{ "" " and " "" } 1 \ge a$

So both $a$ and $b$ must be less than or equal to 1. But, we know that they are both positive integers (greater than 0). Therefore, both $a$ and $b$ must be $1$.

The LCM of $1$ and $1$ is $1$, so that is our final answer.