Question

Question #585c4

Answer 1

`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,

`"number " >= " factor of that number"`

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

`ab` is a factor of both `a` and `b`

This means that `a >= ab` and `b >= ab`

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

`a >= ab " "" "and" "" "b >= ab`

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

`a/a >= (ab)/a " "" " and " "" " b/b >= (ab)/b`

`1 >= b " "" " and " "" " 1 >= 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.