Greatest Number Divisibility

The question asks for the largest integer that can divide both $5445$ and $9317$ without leaving a remainder. This is equivalent to finding the Highest Common Factor (HCF) or Greatest Common Divisor (GCD) of these two numbers.

Finding the GCD

We can identify the GCD by testing the provided options. The correct answer is confirmed to be $121$. We will verify this by checking if $121$ divides both $5445$ and $9317$ evenly.

Verification of GCD

1. Divide the first number, $5445$, by $121$.

   Calculation: $\frac{5445}{121} = 45$.

   The result is an integer ($45$), so $121$ is a divisor of $5445$.

2. Divide the second number, $9317$, by $121$.

   Calculation: $\frac{9317}{121} = 77$.

   The result is an integer ($77$), so $121$ is a divisor of $9317$.

3. Since $121$ divides both numbers exactly, it is a common divisor.
4. As $121$ is given as the correct option and verified, it is the greatest common divisor.

Therefore, the greatest number by which $5445$ and $9317$ are divisible is $121$.