GCF/LCM

[Guest]

I can't seem to find a simple clean definition of Greatest Common Factor or Least Common Multiple. How are they different? How can I remember which is which, and how to find them?

Thank you.

[TakingGMAT]

I can explain it using simple exam.

FOR LCM: you have to look for least common multiple. Take numbers 2 and 5
Multiples of 2 are 2,4,6,8,10...20
Multiples of 5 are 5,10,15,20
Now the least common multiple will be 10

FOR GCF: you have to look for greatest common factor. Take numbers 36 and 54.
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
The factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54.
The common factors of 36 and 54 are 1, 2, 3, 6, 9, 18 but the greatest common factor is 18.

[RonPurewal]

I can explain it using simple exam.

FOR LCM: you have to look for least common multiple. Take numbers 2 and 5
Multiples of 2 are 2,4,6,8,10...20
Multiples of 5 are 5,10,15,20
Now the least common multiple will be 10

FOR GCF: you have to look for greatest common factor. Take numbers 36 and 54.
The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
The factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54.
The common factors of 36 and 54 are 1, 2, 3, 6, 9, 18 but the greatest common factor is 18.

this is all correct, and is an excellent conceptual shortcut for small numbers. so, for instance, if you're asked for the lcm of 9 and 15, you can just scratch your head, think about numbers that are multiples of both 9 and 15, and come up with 45.
BUT
it becomes impracticable fairly quickly if you're dealing with large enough numbers, and it can't be applied at all to variables.

so you need a BETTER GENERAL METHOD for finding lcm's and gcf's.

here it is:
* first, break the numbers down into prime factorizations (surprise, surprise - primes again)
* if you want the lcm, then take ALL primes appearing ANYWHERE, even if they aren't common to both/all the numbers, and take the highest powers of those primes that appear.
* if you want the gcf, then take only those primes that are COMMON to both/all the numbers, and take the lowest powers of those primes.

for instance:
consider
700 = (2^2)(5^2)(7)
440 = (2^3)(5)(11)
you really don't want to go through mental lists of the factors / multiples of these numbers. really, you don't. especially the multiples - good god.
instead, use the approach outlined above:
lcm = (2^3)(5^2)(7)(11) = 15400
gcf = (2^2)(5) = 20

if you have expressions with variables, then you do this process in EXACTLY the same way. so, for instance, (x^2)(y^2)(z) and (x^3)(y)(w) would work in precisely the same way as the above example, except for that "2" is now x, "5" is now y, "7" is now z, and "11" is now w.