The greatest prime number less than or equal to 59 is 59 itself. To determine whether 59 is a prime number, we can follow a simple process of checking for factors.

Definition: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Process:
1. Start by dividing 59 by 2, and check if it is divisible evenly. If it is, then 59 is not a prime number. However, 59 is not divisible by 2.
2. Next, we can check if 59 is divisible by any odd numbers up to its square root, which is approximately 7.68. We don't need to check beyond the square root because any factor greater than the square root would pair with a factor smaller than the square root.
3. Check if 59 is divisible evenly by 3. Again, it is not, so we move on to the next odd number.
4. Check if 59 is divisible evenly by 5. Once again, it is not.
5. Finally, check if 59 is divisible evenly by 7. It is not divisible by 7 either.

Since 59 is not divisible evenly by any numbers between 2 and its square root, we can conclude that 59 is a prime number.

Conclusion:
The greatest prime number less than or equal to 59 is indeed 59 itself.

Therefore, the answer is 59.

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