243 s and 1+5

Step 1: Find the prime factorization of each number. For $243$: $243 = 3 \times 81$ $81 = 3 \times 27$ $27 = 3 \times 9$ $9 = 3 \times 3$ So, the prime factorization of $243$ is $3 \times 3 \times 3 \times 3 \times 3 = 3^5$.

For $145$: $145 = 5 \times 29$ Both $5$ and $29$ are prime numbers. So, the prime factorization of $145$ is $5 \times 29$.

Step 2: Identify common prime factors. The prime factors of $243$ are $\{3\}$. The prime factors of $145$ are $\{5, 29\}$. There are no common prime factors between $243$ and $145$.

Step 3: Determine the Greatest Common Divisor (GCD). Since there are no common prime factors, the greatest common divisor of $243$ and $145$ is $1$.

Alternatively, using the Euclidean algorithm: $243 = 1 \cdot 145 + 98$ $145 = 1 \cdot 98 + 47$ $98 = 2 \cdot 47 + 4$ $47 = 11 \cdot 4 + 3$ $4 = 1 \cdot 3 + 1$ $3 = 3 \cdot 1 + 0$ The last non-zero remainder is $1$.

The Greatest Common Divisor (GCD) of $243$ and $145$ is $\boxed{1}$.