Download NOTES: A quick look at LCM and GCF

NOTES: A quick look at LCM and GCF
multiple: the result of multiplying a number by an integer; a times tables row
multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, …
multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, …
least common multiple (LCM): the smallest shared multiple
What’s the LCM of 3 and 4?
multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, …
multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, …
The first few shared multiples are boldfaced. The smallest one is 12.
That’s the LCM.
What’s the LCM good for? It’s the common denominator for adding or
subtracting fractions. Then it’s called the LCD (least common denominator).
2
3
+
8
12
1
We need to use the common denominator of 12 in order to add these.
4
+
3
12
=
11
12
We’ll talk about how this works when we get to fractions.
factor: an integer you can divide a given number by evenly (without remainder)
factors of 12: 1, 2, 3, 4, 6, 12
factors of 18: 1, 2, 3, 6, 9, 18
 When you find multiples, you get numbers equal to or larger than the
original number. You’re multiplying that number by 1, 2, 3, etc. There’s
no limit to the number of multiples.
 When you find factors, you get numbers equal to or smaller than the
original number. You’re dividing up that number evenly. There’s a fixed
number of factors; the smallest is 1, and the largest is the number itself.
o You multiply factors to get the original number (3  4 = 12), but you divide the
original number to get the factors (12 is divisible by 3 and 4).
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Some divisibility hints: First, know your times tables!




If a number is even (if it ends in 0, 2, 4, 6, 8), it’s divisible by 2.
If a number ends in 0, it’s divisible by 10.
If a number ends in 0 or 5, it’s divisible by 5.
If the digits of a number add up to a number divisible by 3, then the number
itself is divisible by 3. [The digits of 567 add up to 18. Since 18 is divisible by 3, so is 567.]
How do you make sure you find all the factors of a number? [rainbow method]
 Start by writing 1 on the left, some space, and then the number.
[Every number is divisible by 1 and by itself, and 1 times any number equals that number.]
1
12
 If the number isn’t divisible by 2, go to the next step.
If the number is divisible by 2, ask: 2 times what equals the number?
Write 2 and its matching factor like this:
1, 2
6, 12
 If the number isn’t divisible by 3, go to the next step.
If the number is divisible by 3, ask: 3 times what equals the number?
Write 3 and its matching factor like this:
1, 2, 3
4, 6, 12
 Keep going until the left and right sides meet with no factors in between.
1, 2, 3, 4, 6, 12
greatest common factor (GCF): the biggest shared factor
What’s the GCF of 12 and 18?
factors of 12: 1, 2, 3, 4, 6, 12
factors of 18: 1, 2, 3, 6, 9, 18
The common factors are boldfaced. The biggest one is 6. That’s the GCF.
What’s the GCF good for? To reduce fractions you need to find factors common
to the numerator and denominator. You’ll save yourself time if you can spot the
largest shared factor instead of reducing the fraction bit by bit.
12 ÷ 6
18 ÷ 6
=
2
3
We’ll talk about how this works when we get to fractions.
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NOTES: The ladder method for LCM and GCF
[advanced]
 Please make sure you’ve worked through the traditional listing method for
least common multiple (LCM) and greatest common factor (GCF) first. The
ladder method is great because it’s quick; but if you don’t practice it enough,
you can easily forget it. It’s important to be able to fall back on the traditional
method, which is easy to remember since it’s grounded in the meaning of
“LCM” and “GCF.”
EXAMPLE: Find the LCM of 24 and 36.
1. Make an “L” bracket around the 2 numbers (sort of like an upside down long
division).
24
36
2. Think of a number you can divide evenly into both numbers, and write it to the
left of the bracket. (If you can think of a big number, you can save yourself a few
steps, but it’s fine to start with a small number, such as 2 or 3.)
24
36
3. As if you’re doing long division upside down, divide each number inside by the
number outside the bracket.
24
36
4. Ask yourself if there’s a number you can divide into both of the results (here 6
and 9). If so, repeat steps 1-3.
24
36
Keep doing that until there’s nothing you can divide into both of the resulting
numbers (here 2 and 3) except 1. Don’t bother doing the 1 bracket.
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5. To find the LCM, enclose all the outside numbers in a big L (for “LCM”), and
then multiply all those enclosed numbers together:
24
36
LCM = 4  3  2  3 = 72
If you want the GCF instead, multiply ONLY the vertical outside numbers (here the
4 and the 3 in red). Remember: those are the factors that we divided by.
GCF: 4  3 = 12
Frequently asked questions (FAQ’s):
 What if I start by dividing both numbers in the example differently, for
example, by 2 or by 6 instead of by 4? Will I get a different answer?
No. You’ll take more or fewer steps, but the answer will be the same.
Try it and see.
 What if there’s nothing besides 1 that divides into both numbers at the first
step, say when finding the LCM of 5 and 12?
If 1 is all you’ve got, use it so you can complete the “L”. (BTW, it won’t
hurt to do a final step with 1 for every problem, but it’s not necessary
since multiplying by 1 doesn’t change anything.)
5
12
LCM = 1  5  12 = 60
 What if I want to find the LCM of 3 numbers instead of 2?
Use the ladder method for 2 of the numbers. Then do the ladder
method again, this time with the 3rd of the original numbers and the
resulting LCM you calculated with the first ladder. (BTW, you can do
this for the GCF of 3 numbers, too.)
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