Lets say the problem at hand is, we want to simplify this fraction by finding the HCF: 81 / 162. You may have spotted that 81 is the HCF for this, but they want your method.
The "Right" method, when followed, will give a correcct result for this problem, but also for any similar problem, even if harder than this particular problem at hand.
The right method involves listing the prime number factors and number of occurrences in both numbers. A prime factoring tool would make this easier, if you have one.
81 is 9 x 9 or 3 x 3 x 3 x 3 or written as 3^4 for factor of 3 with 4 occurrences.
162 is 9 x 9 x 2 or 3 x 3 x 3 x 3 x 2 or just 2 x 3^4
Then select only the primes and number of occurrences that are the same in both lists. They do not have 2 in common, but do have 3 with 4 occurrences in common.
Multiplying these factors and occurrences will give the correct HCF.
Then divide the HCF into each number in the problem to simplify the fraction.
81 / 81 = 1
162 / 81 = 2
The fraction simplifies to 1 / 2.
So much for the "right" method. There may also be sometimes a shortcut for a particular problem at hand. In this case, you may try dividing the larger number by the smaller one to see if it goes an exact number of times. The answer is yes, it does go exactly 2 times. Thus, even without a prime factoring tool, you know that the smaller number of 81 can serve as the Highest Common Factor to simplify this problem.