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

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

30 and 84 Finding the gcf Begin with two numbers. Step 1 – Complete the prime factorization of these numbers. Do not use exponents. 30 2 x 3 x 5 =30 84 2 x 2 x 3 x 7=84

Finding the gcf 2 x 3 x 5=30 2 x 2 x 3 x 7=84 2 x 3 x 5 =30 Step 2 – Write the prime factorizations underneath each other 2 x 3 x 5=30 2 x 2 x 3 x 7=84 Step 3 – Circle each number that appears in both factorizations 2 x 3 x 5 =30 2 x 2 x 3 x 7=84

Finding the gcf 2 x 3 x 5 =30 2 x 2 x 3 x 7=84 2 x 3=6 Note-The second 2 is not circled in the bottom prime factorization because 2 only appears once in the first factorization. 2 x 3 x 5 =30 2 x 2 x 3 x 7=84 Step 4-Multiply the circled numbers. 2 x 3=6 6 IS YOUR GCF FOR 30 AND 84!

Practice Find the GCF of 24 and 96 2 x 2 x 2 x 3=24 Step 1-Prime Factorization. 96 2 x 2 x 2 x 2 x 2 x 3=96 24 2 x 2 x 2 x 3=24 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

Practice 2 x 2 x 2 x 3=24 24 IS YOUR GCF FOR 24 AND 96! Find the GCF of 24 and 96 Step 3-Circle the common numbers. 2 x 2 x 2 x 3=24 2 x 2 x 2 x 2 x 2 x 3=96 Step 4-Multiply the circled numbers. 2 x 2 x 2 x 3=24 24 IS YOUR GCF FOR 24 AND 96!

Practice Find the GCF of 18 and 35 2 x 3 x 3=18 5 x 7=35 Step 1-Prime Factorization. 35 5 x 7=35 18 2 x 3 x 3=18 Step 2-Write the prime factorizations underneath each other. 2 x 3 x 3=18 5 x 7=35

WE DON’T HAVE ANY COMMON NUMBERS! Practice Find the GCF of 18 and 35 Step 3-Circle the common numbers. 2 x 3 x 3=18 5 x 7=35 WE DON’T HAVE ANY COMMON NUMBERS! Note-If the prime factorizations have no numbers in common, the GCF is 1. 1 IS YOUR GCF FOR 18 AND 35!

Practice Find the GCF of 15 and 36 3 x 5=15 2 x 2 x 3 x 3=36 Step 1-Prime Factorization. 36 2 x 2 x 3 x 3=36 15 3 x 5=15 Step 2-Write the prime factorizations underneath each other. 3 x 5=15 2 x 2 x 3 x 3=36

Step 3-Circle the common numbers. Practice Find the GCF of 15 and 36 Step 3-Circle the common numbers. 3 x 5 =15 2 x 2 x 3 x 3=36 Note-If the prime factorizations only have one number in common, that number is your GCF. 3 IS YOUR GCF FOR 15 AND 36!

Finding the gcf for more than 2 numbers 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.

Finding the gcf for more than 2 numbers Begin with the given numbers. 24, 48, and 60 Step 1 – Complete the prime factorization of these numbers. Do not use exponents. 48 2 x 2 x 2 x 2 x 3=48 24 2 x 2 x 2 x 3 =24 60 2 x 2 x 3 x 5=60

Finding the gcf for more than 2 numbers 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 3 = 48 2 x 2 x 3 x 5 = 60 Step 3 – Circle each number that appears in all factorizations 2 x 2 x 2 x 3 = 24 2 x 2 x 2 x 2 x 3 = 48 2 x 2 x 3 x 5 = 60

Finding the gcf for more than 2 numbers Step 4-Multiply the circled numbers. 2 x 2 x 3=12 12 IS YOUR GCF FOR 24, 48, AND 60! Note-If no number appears in all prime factorizations, then GCF is 1.

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

Notes 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.

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

Finding the LCM-basic method Begin with two numbers. 12 and 80 Step 1 – List several multiples of each number. 12 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, 180, 192, 204, 216, 228, 240, 252, 264, 276, 288, 300 80 80, 160, 240, 320, 400

Finding the LCM-basic method Step 2-Find the smallest number that occurs in both lists. 12 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, 180, 192, 204, 216, 228, 240, 252, 264, 276, 288, 300 80 80, 160, 240, 320, 400 240 is you LCM for 12 and 80! Note-You will always have to list more multiples of the smaller number.

208 is your LCM for 16 and 52 practice 16 52 Find the LCM of 16 and 52 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240, 256, 272 52 52, 104, 156, 208, 260, 312 208 is your LCM for 16 and 52

323 is your LCM for 17 and 19 practice 17 19 Find the LCM of 17 and 19 17, 34, 51, 68, 85, 102, 119, 136, 153, 170, 187, 204, 221, 238, 255, 272, 289 306, 323, 340, 357, 374, 391, 408, 425 19 19, 38, 57, 76, 95, 114, 133, 152, 171, 190, 209, 228, 247, 266, 285, 304, 323, 342, 361, 380, 399, 418, 437, 456 323 is your LCM for 17 and 19 Note-The LCM of two prime numbers is always the product of the numbers.

Finding the LCM for more than 2 numbers-basic method 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.

Finding the LCM for more than 2 numbers-basic method Find the LCM of 8, 30, and 48 8 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 48 48, 96, 144, 192, 240, 288, 336, 384, 432, 480 30 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330 240 is your LCM for 8, 30, and 48

Finding the LCM-advanced method Begin with two numbers. 12 and 80 Step 1 – Complete the prime factorization of these numbers. Do not use exponents. 12 2 x 2 x 3 =12 80 2 x 2 x 2 x 2 x 5=80

Finding the LCM-advanced method Step 2 – Write the prime factorizations underneath each other 2 x 2 x 3=12 2 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=12 2 x 2 x 2 x 2 x 5= 80 2 appears twice, 3 appears once 2 appears four times, 5 appears once

Finding the LCM-advanced method Step 4-Use the highest occurrence of each number to create a multiplication problem. 2 x 2 x 3=12 2 x 2 x 2 x 2 x 5= 80 2 appears twice, 3 appears once 2 appears four times, 5 appears once 2 x 2 x 2 x 2 x 3 x 5= 5 appears once in the second factorization 2 appears four times in the second factorization 3 appears once in the first factorization

Finding the LCM-advanced 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!

practice 12 26 Find the LCM of 12 and 26 2 x 2 x 3 =12 2 x 13 =26 Step 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 =12 2 x 13 =26

2 x 2 x 3 x 13= practice Find the LCM of 12 and 26 2 x 2 x 3 =12 Step 3 – Count the times each factor appears in each factorization. 2 x 2 x 3 =12 2 x 13 =26 2 appears twice, 3 appears once 2 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 13 appears once in the second factorization 3 appears once in the first factorization

Finding the LCM-advanced Step 5 – Solve the problem 2 x 2 x 3 x 13= 156 156 is you LCM for 12 and 26!

Notes 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.

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