Question

The greatest common factor (GCF) of x^3, x^7, x^9 is ?
Please help

Answer 1

`x^3`

Explanation:

Let the greatest common factor of
`a,b,c` such that
`a,b,c \in\mathbb{Z}` and they are not all equal to zero,
`d` is the common divisor of
`a` and
`b`. Therefore,
`d \mid a` and
`d \mid b`

You can write

`D(a) = \{d \in\mathbb{Z} : d \mid a\}`

`D(b) = \{d \in\mathbb{Z} : d \mid b\}`

The greatest common factor of
`a,b` is given as

`D(a,b) = \{d \in\mathbb{Z} : d \mid a \text{ and } d \mid b\}`

and

`D(a, b) = D(a) \cap D(b)`

This happens because
`D(a,b)` is upper bounded because if
`a\\eq 0` then
`d \leq |a|`

for all
`d\in D(a, b)`. Therefore, the set
`D(a, b)` has the greatest elements.

Taking
`x^3, x^7, x^9` such that
`x>0`

You can note that
`x^7 = x^3 \cdot x^4` and
`x^9 = x^3 \cdot x^6 = x^3 \cdot x^3 \cdot x^3`

Therefore, the greatest common factor is
`x^3`

Note:
`x^3 \cap x^7 \cap x^9 = x^3`