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Why wouldn't you be asked to find the greatest common factor?
Because no one likes a show-off.
Why would 2 be the greatest common factor in a problem?
Because the numbers being asked about are consecutive even numbers.
What is the greatest common factor of 73 and 127?
This is tricky. Once you determine that 73 is a prime number then you can ignore 127 (since 73 is not a factor of 127). So, the greatest common factor is 1. Note: if you asked 73 and 146 then the answer would be 73. Anytime one of the numbers is a prime - then the answer is EITHER 1 or the prime number (if the other number is a multiple of the prime).
What is the least common multiple of 65?
When taking the LCM you need two numbers. the first 6 multiples of 65 are 65, 130, 195, 260, 325,390 to find more multiples just multiply 65 by 7 then 8 then 9 then 10 then 11... etc... Now you asked to find the Least Common Multiple. You can not find something in common if you do not have something the compare it to.
How is common factors and greatest common factors similar and different?
They really aren't all that similar... Greastest common factor finds the greatest number that you can divide both numbers by... for example, the GCF of 15 and 20 is 5, because 15 / 3 = 5 and 20 / 4 = 5, there is no higher number for division purposes that the two share. 10 is higher than 5 in the case of 20, but you cannot divide 15 by 10 without getting a remainder of 5. Least common multiple finds the smallest common number that you can get when multiplying a number... for example, the LCM of 15 and 45 is 15 itself; 15 x 1 = 15, and 15 x 3 = 45. There is no smaller number that you can find that is common between the two. Another example, since the previous one had an LCM of an actual number being asked would be 18 and 24... in this case it would be 72 because there is no smaller number that the two share when multiplied by another value. They are somewhat similar in how you find the end value, however. By dividing the two given numbers in the case of LCM, you will eventually break them up into prime numbers. Eliminating all similar numbers and then multiplying what is left will give you the LCM. In the case of GCF, by doing the same thing minus multiplying the prime numbers, you will end up finding the greatest number that the two are divisible by.
