Answer:
[tex]x^3[/tex]
Step-by-step explanation:
Let the greatest common factor of [tex]a,b,c[/tex] such that [tex]a,b,c \in\mathbb{Z}[/tex] and they are not all equal to zero, [tex]d[/tex] is the common divisor of [tex]a[/tex] and [tex]b[/tex]. Therefore, [tex]d \mid a[/tex] and [tex]d \mid b[/tex]
You can write
[tex]D(a) = \{d \in\mathbb{Z} : d \mid a\}[/tex]
[tex]D(b) = \{d \in\mathbb{Z} : d \mid b\}[/tex]
The greatest common factor of [tex]a,b[/tex] is given as
[tex]D(a,b) = \{d \in\mathbb{Z} : d \mid a \text{ and } d \mid b\}[/tex]
and
[tex]D(a, b) = D(a) \cap D(b)[/tex]
This happens because [tex]D(a,b)[/tex] is upper bounded because if [tex]a\neq 0[/tex] then [tex]d \leq |a|[/tex]
for all [tex]d\in D(a, b)[/tex]. Therefore, the set [tex]D(a, b)[/tex] has the greatest elements.
Taking [tex]x^3, x^7, x^9[/tex] such that [tex]x>0[/tex]
You can note that [tex]x^7 = x^3 \cdot x^4[/tex] and [tex]x^9 = x^3 \cdot x^6 = x^3 \cdot x^3 \cdot x^3[/tex]
Therefore, the greatest common factor is [tex]x^3[/tex]
Note: [tex]x^3 \cap x^7 \cap x^9 = x^3[/tex]
