The greatest prime factor of 308 is 11.
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The greatest prime factor of 40 is: 5
11 is the greatest prime factor of 99.
The greatest prime factor of both 6 and 27 is 3.
Considering that 100 = (2^2)*(5^5) and 38 = (2^1)*(19^1), we can see that 2 is the greatest common prime factor between 100 and 38. If the question meant the greatest prime factor of either 100 or 38, then the answer is 19.
You do a factor rainbow to find a prime factorization. You compare prime factorizations to find a greatest common factor.