NOTES: A quick look at LCM and GCF
NOTES: A quick look at LCM and GCF What’s a multiple? the result of multiplying a number by an integer; a times tables row EXAMPLES: multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, … multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, …
What’s the least common multiple (LCM)? the smallest shared multiple EXAMPLE: 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.
When do you use LCM? It’s the most efficient common denominator for adding or subtracting fractions. Then it’s called the LCD (least common denominator).
2 9
+
1 6
The LCM of 9 and 6 is 18. We can use the least common denominator of 18 in order to add these fractions.
+
2
2
9
9
1 6
+
1 6
= =
2 18 18
9
+
1 6
= =
4 18 3 18 7 18
© 2017 D. Stark
10/19/2017
OVERVIEW: LCM and GCF
1
You can add/subtract fractions by using the “easy common denominator” instead. To find that you just multiply the denominators. (In the example above, it would be 9 6 = 54.) But then you’ll have to reduce at the end. If you use the least common denominator, you won’t have to do so much reducing.
What’s a factor? an integer you can divide a given number by evenly (without remainder) EXAMPLES: factors of 12: 1, 2, 3, 4, 6, 12 factors of 18: 1, 2, 3, 6, 9, 18
NOTE on multiples & factors: 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. You multiply factors to get the original number (3 4 = 12), but you divide the originaldo number get the factorsa(12 is divisible by and 4). How youto tell what number is3divisible by?
If a number ends in 25, it’s divisible by 25. 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 a number is even (if it ends in 0, 2, 4, 6, 8), it’s divisible by 2. 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.] After that, use your times tables knowledge to spot factors.
© 2017 D. Stark
10/19/2017
OVERVIEW: LCM and GCF
2
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
What’s the greatest common factor (GCF)? the biggest shared factor EXAMPLE: 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.
When do you use the GCF? 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 12 ÷ 6 2 factor instead of reducing the fraction bit by bit. = 18 ÷ 6 3 Also in algebra you’ll use the GCF when factoring polynomials: 12x + 18 = 6(x + 18). © 2017 D. Stark
10/19/2017
OVERVIEW: LCM and GCF
3
What’s 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.
© 2017 D. Stark
10/19/2017
OVERVIEW: LCM and GCF
4
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) on the ladder method: 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.) © 2017 D. Stark
10/19/2017
OVERVIEW: LCM and GCF
5
What’s the least common multiple (LCM)? the smallest shared multiple EXAMPLE: 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.
When do you use LCM? It’s the most efficient common denominator for adding or subtracting fractions. Then it’s called the LCD (least common denominator).
2 9
+
1 6
The LCM of 9 and 6 is 18. We can use the least common denominator of 18 in order to add these fractions.
+
2
2
9
9
1 6
+
1 6
= =
2 18 18
9
+
1 6
= =
4 18 3 18 7 18
© 2017 D. Stark
10/19/2017
OVERVIEW: LCM and GCF
1
You can add/subtract fractions by using the “easy common denominator” instead. To find that you just multiply the denominators. (In the example above, it would be 9 6 = 54.) But then you’ll have to reduce at the end. If you use the least common denominator, you won’t have to do so much reducing.
What’s a factor? an integer you can divide a given number by evenly (without remainder) EXAMPLES: factors of 12: 1, 2, 3, 4, 6, 12 factors of 18: 1, 2, 3, 6, 9, 18
NOTE on multiples & factors: 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. You multiply factors to get the original number (3 4 = 12), but you divide the originaldo number get the factorsa(12 is divisible by and 4). How youto tell what number is3divisible by?
If a number ends in 25, it’s divisible by 25. 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 a number is even (if it ends in 0, 2, 4, 6, 8), it’s divisible by 2. 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.] After that, use your times tables knowledge to spot factors.
© 2017 D. Stark
10/19/2017
OVERVIEW: LCM and GCF
2
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
What’s the greatest common factor (GCF)? the biggest shared factor EXAMPLE: 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.
When do you use the GCF? 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 12 ÷ 6 2 factor instead of reducing the fraction bit by bit. = 18 ÷ 6 3 Also in algebra you’ll use the GCF when factoring polynomials: 12x + 18 = 6(x + 18). © 2017 D. Stark
10/19/2017
OVERVIEW: LCM and GCF
3
What’s 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.
© 2017 D. Stark
10/19/2017
OVERVIEW: LCM and GCF
4
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) on the ladder method: 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.) © 2017 D. Stark
10/19/2017
OVERVIEW: LCM and GCF
5
