To test if 6637 we have to start by calculating the root of it:
![\sqrt[]{6637}=81.46\ldots](https://img.qammunity.org/qa-images/2023/formulas/mathematics/college/5u7wba0hczqdd0euqdt2.png)
The nearest prime factor to the root of 6637 is 79.
We now have to list all the prime numbers that are smaller or equal than 79 and then, if none of them are factors of 6637, then 6637 is a prime.
The list is: 2, 3, 5, 7, 11, 13, 17, 23, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79.
We test if any of this prime numbers is a factor of 6637. None of them are factors of 6637, then 6637 is a prime number.
We test all the prime numbers as factors up to 79.
Then, the greatest prime to consider is 79, that is the greatest prime that is smaller than the square root of the number.
Answer: the greatest prime to consider is 79.