Pre Algebra Period 1

Comparing and Ordering Fractions

 5-1
HOW TO FIND THE LEAST COMMON MULTIPLE:

LCM = Smallest number that your numbers can go into.
Just like GCF, let's look at the letters backwards to understand it!

Multiple = each number given in the problem must go into this number (example for multiples of 2: 2, 4, 6, 8.... multiples of 3: 3, 6, 9, 12...     multiples of 5: 5, 10, 15, 20...)

Common = must be a number that ALL THE NUMBERS go into 

Least = must be the SMALLEST number that ALL THE NUMBERS go into

There are the same ways to find it as the GCF:

1) List multiples of each number and circle the smallest one that is common to all the numbers
---> most of the time takes WAY TOO long!!
2) Circle every factor in the prime factorizations of each number that is different and multiply

3) List the EXPANDED FORM prime factorizations in a table and bring down ONE OF EACH COLUMN. Then multiply. (or you can do this with exponential form but you need to bring down the HIGHEST POWER of each column).
4) Using the BOX method from class create an L from the left side and the bottom row of relatively prime factors. It is their product.

THE DIFFERENCE BETWEEN GCF AND LCM:
For the GCF, you need the LEAST POWER of only the COMMON FACTORS.
For the LCM, you need the GREATEST POWER of EVERY FACTOR.

WHY DO WE NEED EVERY FACTOR THIS TIME?

Because it's a multiple of all your numbers!

Multiples start with each number, so all the factors that make up each number have to be in this common multiple of all the numbers.

For example, say we're finding the LCM of 12 and 15, that multiple must be a multiple of 12: 12, 24, etc.
AND 15: 15, 30, etc.
So the COMMON multiple must include 12 (2x2x3) AND 15 (3x5).
The LCM must have two 2s and one 3 or 12 won't go into it.
It must also have that same 3 that 12 needs and one 5 or it won't be a multiple of 15.
EXAMPLE: 54 and 36

LIST MULTIPLES OF EACH NUMBER UNTIL YOU SEE ONE THAT BOTH OF THEM GO INTO (the LEAST MULTIPLE that is COMMON to both)
36 = 36, 72, 108, 144 ...
54 = 54, 108
This more difficult than the listing method for GCF for 2 reasons: You don't know where to stop as you least the first number...multiples go on forever! and the numbers get big very fast because they're MULTIPLES, not factors!
PRIME FACTORIZATION METHOD
54 = 2x3x3x3

36 = 2x2x3x3
The LCM will be one of each factor of each number (but don't double count a factor that is common to both numbers...once you have it in your LCM, check it off in the other number if it's common...you don't need that factor again)

LCM = 2x2x3x3x3 = 108
Let's look at this calculation and think about why it works:
Why does it need TWO 2s?
Although 54 only needs ONE, 36 needs TWO or 36 won't go into the LCM.
Try it putting in only ONE 2: 2x3x3x3 = 54. Does 36 go into 54? NO
Why does the LCM need THREE 3s? Although 36 needs only TWO 3s, 54 needs THREE. Try it putting in only TWO: 2x2x3x3 = 36.
36 is TOO SMALL to even be a MULTIPLE of 54!
WHAT IF BY MISTAKE YOU "DOUBLE" UP FACTORS AND USE ALL OF THEM?
For the GCF, you would have got a number way too big to be a factor of the numbers. For the LCM, you will STILL GET A COMMON MULTIPLE! But it WON'T BE THE LEAST!
In fact if you double up, generally you'll get a really big number and the bigger the number is, the harder it is to use.
For example, if you use all the factors of both 36 and 54, you're just multiplying 36 x 54 = ???? 
1944!!!! 
Would you rather use 108 or 1944???


LCM WITH A COLUMN APPROACH:
You place the prime factorization for each number in columns like we did for the GCF, matching the factors in each column.
If a factor doesn't match, it gets a separate column.
YOU'LL JUST TAKE ONE OF EACH COLUMN AND MULTIPLY!

                                    SAME EXAMPLE: 54, 36
                                     
36 = 2 x 2 x 3 x 3
                                     
54 = 2 x       3 x 3 x 3
                                   
LCM = 2 x 2 x 3 x 3 x 3
= 108
You can do it in exponential form, too.

If you do it in exponential form, you take the HIGHEST POWER of each column!

Works with variables the exact same way!
Take the HIGHEST power of EVERY variable (not just the ones that are common like the GCF)
YOU CAN FIND BOTH THE GCF AND THE LCM IN THIS SAME COLUMN FORMAT!

CHECKING THE LCM TO MAKE SURE IT WORKS: