### Instruction: The symbol $5!$ means $5\cdot 4\cdot 3\cdot 2\cdot 1$. What is the greatest odd integer that is a factor of $5!$ ?### Response: I know that $5!$ is the product of all the positive integers from $1$ to $5$, so any factor of $5!$ must be a product of some subset of these numbers.
I also know that any odd factor of $5!$ must not include any even numbers in its product, since an even number times any other number is still even.
Therefore, the greatest odd factor of $5!$ must be the product of all the odd numbers from $1$ to $5$, which are $1, 3, 5$.
So the answer is $1\cdot 3\cdot 5 = 15$.