Figuring Out The Highest Common Factor
So you need the highest common factor of 24 and 36. Here is how it actually works in practice, not the textbook version. Start by breaking both numbers into their prime factors. That is the method most people overlook, and it saves you from making silly mistakes later. 24 breaks down to 2 times 2 times 2 times 3. 36 breaks down to 2 times 2 times 3 times 3. Now look for what they share. Both have two 2s and one 3. Multiply those together and you get 12. That is your answer. Pretty straightforward once you see it laid out. I remember dealing with a situation years ago where I had to find the HCF of much larger numbers on a construction site—someone needed to tile a floor with two different sizes of tiles and figure out the largest square tile that would fit evenly into both dimensions without cutting anything. The numbers were something like 1,248 and 1,872. Hand- factoring those would have been a nightmare. I just ran the Euclidean algorithm instead: divide the bigger by the smaller, take the remainder, repeat until the remainder hits zero. Last non-zero remainder is your HCF. Took about 30 seconds.
For smaller numbers like 24 and 36, prime factorization is perfectly fine. But there is a catch. If the numbers are big enough that factoring becomes tedious, you are better off switching methods. The prime factorization approach also falls apart when you are working with variables or algebraic expressions where you can't just "factor" in the traditional sense. In those cases the Euclidean algorithm is your fallback. Another thing people mess up is confusing the HCF with the LCM. They are related but completely different. The HCF gives you the largest number that divides evenly into both. The LCM gives you the smallest number that both divide into. For 24 and 36 the LCM is 72. You can get there quickly by multiplying the two numbers and dividing by the HCF: 24 times 36 divided by 12 equals 72. Useful shortcut if you need both at the same time. The prime factorization method has a real limitation though. It does not scale well. Factoring a number like 1,001 by hand requires you to know your divisibility rules cold, and even then you might waste five minutes on trial division. The Euclidean algorithm handles that in three steps. That is not opinion—that is just arithmetic.
Also worth noting: some people try to find the HCF by listing all the factors. For 24 that means writing out 1, 2, 3, 4, 6, 8, 12, 24. For 36 it is 1, 2, 3, 4, 6, 9, 12, 18, 36. Then you pick the biggest match. It works for small numbers. It gets ridiculous fast. Don't do this unless you are teaching a sixth grader. Bottom line: the highest common factor of 24 and 36 is 12. The method you choose depends on the size of the numbers and whether you are doing this by hand or on a machine. For everyday use with small integers, prime factorization is clear enough. For anything larger or more complex, go with the Euclidean algorithm.
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