Greatest common factors ( gcf ) & least common multiples (lcm)

• Greatest Common Factor (GCF) – The largest number that divides exactly into two or more numbers

30 and 84Begin with two numbers.

Step 1 – Complete the prime factorizationof these numbers. Do not use exponents.

30

2 x 3 x 5 =30

84

2 x 2 x 3 x 7=84

Step 2 – Write the prime factorizations underneath each other

2 x 3 x 5=302 x 2 x 3 x 7=84

Step 3 – Circle each number that appears in both factorizations

2 x 3 x 5 =302 x 2 x 3 x 7=84

Note-The second 2 is not circled in the bottom prime factorizationbecause 2 only appears once in the first factorization.

2 x 3 x 5 =302 x 2 x 3 x 7=84

Step 4-Multiply the circled numbers.

2 x 3=66 IS YOUR GCF FOR 30 AND 84!

Find the GCF of 24 and 96Step 1-Prime Factorization.

24

2 x 2 x 2 x 3=24

96

2 x 2 x 2 x 2 x 2 x 3=96

Step 2-Write the prime factorizations underneath each other.

2 x 2 x 2 x 3=24

2 x 2 x 2 x 2 x 2 x 3=96

Find the GCF of 24 and 96Step 3-Circle the common numbers.

Step 4-Multiply the circled numbers.

2 x 2 x 2 x 3=24

2 x 2 x 2 x 2 x 2 x 3=96

2 x 2 x 2 x 3=2424 IS YOUR GCF FOR 24 AND 96!

Find the GCF of 18 and 35Step 1-Prime Factorization.

18

2 x 3 x 3=18

35

5 x 7=35

Step 2-Write the prime factorizations underneath each other.

2 x 3 x 3=18

5 x 7=35

Find the GCF of 18 and 35Step 3-Circle the common numbers.

Note-If the prime factorizations have no numbers in common,the GCF is 1.

1 IS YOUR GCF FOR 18 AND 35!

2 x 3 x 3=18

5 x 7=35WE DON’T HAVE ANY COMMON NUMBERS!

Find the GCF of 15 and 36Step 1-Prime Factorization.

15

3 x 5=15

36

2 x 2 x 3 x 3=36

Step 2-Write the prime factorizations underneath each other.

3 x 5=15

2 x 2 x 3 x 3=36

Find the GCF of 15 and 36Step 3-Circle the common numbers.

3 x 5 =15

2 x 2 x 3 x 3=36

3 IS YOUR GCF FOR 15 AND 36!

Note-If the prime factorizations only have one number in common, that number is your GCF.

• To find the GCF for more than 2 numbers, take the same steps as before, but you will circle the common numbers in all of the prime factorizations.

24, 48, and 60Begin with the given numbers.

Step 1 – Complete the prime factorizationof these numbers. Do not use exponents.

24

2 x 2 x 2 x 3 =24

48

2 x 2 x 2 x 2 x 3=48

60

2 x 2 x 3 x 5=60

Step 2 – Write the prime factorizations underneath each other

Step 3 – Circle each number that appears in all factorizations

2 x 2 x 2 x 3 = 242 x 2 x 2 x 2 x 3 = 482 x 2 x 3 x 5 = 60

2 x 2 x 2 x 3 = 242 x 2 x 2 x 2 x 3 = 482 x 2 x 3 x 5 = 60

Step 4-Multiply the circled numbers.

2 x 2 x 3=1212 IS YOUR GCF FOR 24, 48, AND 60!

Note-If no number appears in all prime factorizations, then GCF is 1.

• To solve this problem, you could do all of the prime factorizations for the numbers listed. However, 31 is a prime number. Whenever a prime number is in your given set of numbers, and all the other numbers are not multiples of the prime number, then the GCF is always 1.

Find the GCF of 12, 16, 31, 44, 68, and 96.

• Carefully check that you have circled all of the matching prime factors and haven’t circled any non-matching ones.

• If the set of given numbers does not have any prime factors in common, the GCF is 1.

• If the set of given numbers only has one prime factor in common, that prime factor is your GCF.

• To find the GCF for more than two numbers, the prime factors must be present in all prime factorizations.

• If one (or more) of the numbers in the given set is a prime number, and all of the other numbers are not multiples of the prime number, the GCF is always 1.

Questions?

• Least Common Multiple (LCM) – The smallest number that is a multiple of two or more numbers.

12 and 80Begin with two numbers.

Step 1 – List several multiples of each number.

1212, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, 180, 192, 204, 216,

228, 240, 252, 264, 276, 288, 300

8080, 160, 240, 320, 400

1212, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, 180, 192, 204, 216,

228, 240, 252, 264, 276, 288, 300

8080, 160, 240, 320, 400

Step 2-Find the smallest number that occurs in both lists.

240 is you LCM for 12 and 80!Note-You will always have to list more multiples of the smaller number.

Find the LCM of 16 and 52

1616, 32, 48, 64, 80, 96, 112, 128, 144,

160, 176, 192, 208, 224, 240, 256, 272

5252, 104, 156, 208, 260, 312

208 is your LCM for 16 and 52

Find the LCM of 17 and 19

1717, 34, 51, 68, 85, 102, 119, 136, 153, 170, 187, 204, 221, 238, 255, 272, 289306, 323, 340, 357, 374, 391, 408, 425

323 is your LCM for 17 and 19

1919, 38, 57, 76, 95, 114, 133, 152, 171,

190, 209, 228, 247, 266, 285, 304, 323,342, 361, 380, 399, 418, 437, 456

Note-The LCM of two prime numbers is always the product of the numbers.

• To find the LCM for more than 2 numbers, take the same steps as before, but you will circle the first multiple that is present in all lists.

Find the LCM of 8, 30, and 488

8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 144, 152

160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280

3030, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330

4848, 96, 144, 192, 240,

288, 336, 384, 432, 480

240 is your LCM for 8, 30, and 48

12 and 80Begin with two numbers.

Step 1 – Complete the prime factorizationof these numbers. Do not use exponents.

12

2 x 2 x 3 =12

80

2 x 2 x 2 x 2 x 5=80

Step 2 – Write the prime factorizations underneath each other

2 x 2 x 3=122 x 2 x 2 x 2 x 5= 80

Step 3 – Count the times each factor appears in each factorization.

2 x 2 x 3=122 x 2 x 2 x 2 x 5= 80

2 appears twice, 3 appears once2 appears four times, 5 appears once

Step 4-Use the highest occurrence of each number to create a multiplication problem.

2 x 2 x 3=122 x 2 x 2 x 2 x 5= 80

2 appears twice, 3 appears once2 appears four times, 5 appears once

2 x 2 x 2 x 2 x 3 x 5=

2 appears four times in the second factorization

3 appears once in the first factorization

5 appears once in the second factorization

Step 5 – Solve the problem

2 x 2 x 2 x 2 x 3 x 5= 240

240 is you LCM for 12 and 80!

Find the LCM of 12 and 26Step 1 – Prime factorization

12

2 x 2 x 3 =12

26

2 x 13 =26

Step 1 – Write the prime factorizations underneath each other.

2 x 2 x 3 =122 x 13 =26

Step 3 – Count the times each factor appears in each factorization.

Find the LCM of 12 and 26

2 x 2 x 3 =122 x 13 =26

2 appears twice, 3 appears once2 appears once, 13 appears once

Step 4-Use the highest occurrence of each number to create a multiplication problem.

2 x 2 x 3 x 13=2 appears twice in the first factorization 3 appears once in the

first factorization13 appears once in the second factorization

Step 5 – Solve the problem

2 x 2 x 3 x 13=156

156 is you LCM for 12 and 26!

The LCM of two prime numbers is always the product of the two numbers.

The basic method is better for smaller numbers.

The advanced method is better for larger numbers.

Questions?