Find the greatest common factor of $144$ and $405$.
To find the prime factorization of $144$, I can start by dividing by $2$ repeatedly until I get an odd number. $144 = 2 \cdot 72 = 2 \cdot 2 \cdot 36 = 2 \cdot 2 \cdot 2 \cdot 18 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 9 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 = 2^4 \cdot 3^2$. To find the prime factorization of $405$, I can start by dividing by $3$ repeatedly until I get a prime number. $405 = 3 \cdot 135 = 3 \cdot 3 \cdot 45 = 3 \cdot 3 \cdot 3 \cdot 15 = 3 \cdot 3 \cdot 3 \cdot 3 \cdot 5 = 3^4 \cdot 5$. Now I can compare the prime factorizations of $144$ and $405$ and find the common prime factors with the lowest exponents. They have $3$ as a common prime factor, and the lowest exponent is $2$. Therefore, the greatest common factor of $144$ and $405$ is the product of the common prime factors with the lowest exponents, which is $3^2 = 9$. 