Common divisors

$10800={ 2 }^{ 4 }{ \cdot 3 }^{ 3 }\cdot { 5 }^{ 2 }$

$9000={ 2 }^{ 3 }{ \cdot 3 }^{ 2 }\cdot { 5 }^{ 3 }$

The common prime factors of $10800$ and $9000$ are ${ 2 }^{ 3 }\cdot { 3 }^{ 2 }{ \cdot 5 }^{ 2 }$ . So the common divisors are factors of ${ 2 }^{ 3 }\cdot { 3 }^{ 2 }{ \cdot 5 }^{ 2 }$ .

Their number is equal to $\left( 3+1 \right) \cdot \left( 2+1 \right) \cdot \left( 2+1 \right) =36$