GREATEST COMMON FACTOR AND LEAST COMMON MULTIPLE
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1) Greatest common factors (GCF) and least common multiples (LCM) are important concepts in mathematics. The GCF is the largest integer that divides two or more numbers, while the LCM is the smallest integer that is a multiple of two or more numbers. 2) There are several methods to find the GCF and LCM, including listing factors, using a factor tree, and continuous division. These methods involve breaking numbers down into their prime factors to identify common factors or multiples. 3) For the example of finding the GCF and LCM of 36 and 48, the GCF is 12 and the LCM is 144 using either the factor tree or continuous division methods.
GREATEST COMMON FACTOR AND LEAST COMMON MULTIPLE
- 1. Greatest Common Factors and Least Common Multiples
- 3. Factors of a Number A number may be made by multiplying two or more other numbers together.
- 4. Factors of a Number Factors of 24 1 X 24 2 X 12 3 X 8 4 X 6 {1, 2, 3, 4, 6, 8, 12, 24} A list of its factors Multiplication combinations to reach 24
- 5. Greatest Common Factor (GCF) Greatest common factor is the greatest or biggest whole number that is a factor of two or more given whole numbers. next page
- 6. To find the Greatest Common Factor of a number you can using PRIME FACTORIZATION: Listing Method Factor Tree Continuous Division
- 7. What is the greatest common factor of 36 and 48?
- 9. LISTING METHOD is a process in which you write down all the factors of the given numbers.
- 10. What is the greatest common factor of 36 and 48? Factors of 36: Factors of 48: {1, 2, 3, 4, 6, 9, 12, 18, 36} 36 1 2 18 3 12 4 9 6 {1, 2, 3, 4, 6, 8,12, 16, 24, 48}
- 11. Factors of 36: Factors of 48: {1, 2, 3, 4, 6, 9, 12, 18, 36} {1, 2, 3, 4, 6, 8,12, 16, 24, 48} What is the greatest common factor of 36 and 48? Which factors are common to both numbers? 12 12 6 6 4 4 3 3 2 2 1 1
- 12. Factors of 36: Factors of 48: {1, 2, 3, 4, 6, 9, 12, 18, 36} {1, 2, 3, 4, 6, 8,12, 16, 24, 48} What is the greatest common factor of 36 and 48? 12 12 6 6 4 4 3 3 2 2 1 1 Which is the greatest common factor?
- 14. FACTOR TREE METHOD is a diagrams that show the prime factors of numbers in a branching manner.
- 15. - - Factor Tree Method - - 2 x 18 2 x 9 3 x 3 36 48
- 16. - - Factor Tree Method - - 2 x 18 2 x 9 3 x 3 48 2 x 24 2 x 12 2 x 6 36 2 x 3
- 17. - - Factor Tree Method - - 36 = 2 x 2 x 3 x 3 48 = 2 x 2 x 2 x 2 x 3 2 x 18 2 x 9 3 x 3 48 2 x 24 2 x 12 2 x 6 36 2 x 3
- 18. - - Factor Tree Method - - 36 = 2 x 2 x 3 x 3 48 = 2 x 2 x 2 x 2 x 3 Now that the numbers are written as a product of their prime factors. Identify the common prime factors then multiply it all the product is the GCF.
- 19. - - Factor Tree Method - - 36 = 2 x 2 x 3 x 3 48 = 2 x 2 x 2 x 2 x 3 GCF= 2 x 2 x 3 GCF= 12
- 21. CONTINUOUS DIVISION is the process of dividing a number by possible prime factors until there is no other number besides 1 that is a factor of both numbers.
- 22. CONTINUOUS DIVISION 36 48 Think of any number that divides evenly into both 36 and 48 (a factor of both) For this example, I’ll use 2. 2 18 24
- 23. 36 48 2 18 24 CONTINUOUS DIVISION Think of any number that divides evenly into both 18 and 24 (a factor of both) For this example, I’ll use 3. 3 Divide both numbers by 3. 6 8 2
- 24. 36 48 2 18 24 CONTINUOUS DIVISION Continue this process until there is no other number besides 1 that is a factor of both numbers. 3 3 and 4 have no common factor except 1... so you can stop dividing. 6 8 2 2 3 4
- 25. CONTINUOUS DIVISION 18 24 3 6 8 2 2 3 4 36 48 The product of the numbers along the LEFT SIDE is the GCF of 36 and 48. GCF= 2 x 3 x 2 GCF = 12
- 27. Multiples of a Number The products of a number with the natural numbers 1, 2, 3, 4, 5, ... are called the multiples of the number. For example: 7 x 1 = 7 7 x 2 = 14 7 x 3 = 21 7 x 4 = 28 So, the multiples of 7 are 7, 14, 21, 28, and so on.
- 28. Further examples: multiples of 2 are 2, 4, 6, 8, … multiples of 3 are 3, 6, 9, 12, … multiples of 4 are 4, 8, 12, 16, … Multiples of a Number
- 29. Least Common Multiple (LCM) The smallest common multiple of two or more numbers is called the Least Common Multiple (LCM).
- 30. To find the LEAST COMMON MULTIPLE of a number you can using PRIME FACTORIZATION: Listing Method Factor Tree Continuous Division
- 31. LISTING METHOD
- 32. LISTING METHOD To find the least common multiple (LCM) of two or more numbers, list the multiples of the larger number and stop when you find a multiple of the other number. This is the LCM.
- 33. Least Common Multiple (LCM) Find the least common multiple of 6 and 9. Solution: List the multiples of 9 and stop when you find a multiple of 6. Multiples of 9 are 9, 18, … Multiples of 6 are 6, 12, 18, … LCM is 18
- 34. What is the Least Common Multiple of 36 and 48?
- 35. Least Common Multiple (LCM) Find the least common multiple of 36 and 48. Solution: List the multiples of 36 and stop when you find a multiple of 48. 36 = 36,72, 108, 144, 180, … 48 = 48, 96, 144, … LCM is 144
- 36. Prime Factorization to find LCM With the Factor Tree and Continuous Division methods, you can find both the GCF and LCM at the same time! Let’s use a previous example…
- 37. Find the least common multiple of 36 and 48.
- 39. - - Factor Tree Method - - 2 x 18 2 x 9 3 x 3 36 48
- 40. - - Factor Tree Method - - 2 x 18 2 x 9 3 x 3 48 2 x 24 2 x 12 2 x 6 36 2 x 3
- 41. - - Factor Tree Method - - 36 = 2 x 2 x 3 x 3 48 = 2 x 2 x 2 x 2 x 3 2 x 18 2 x 9 3 x 3 48 2 x 24 2 x 12 2 x 6 36 2 x 3
- 42. - - Factor Tree Method - - 36 = 2 x 2 x 3 x 3 48 = 2 x 2 x 2 x 2 x 3 Now that the numbers are written as a product of their prime factors. Multiply all the prime factors. Take note, common factor should be written once only.
- 43. - - Factor Tree Method - - 36 = 2 x 2 x 3 x 3 48 = 2 x 2 x 2 x 2 x 3 LCM = 2 x 2 x 3 x 2 x 2 x 3
- 44. - - Factor Tree Method - - LCM = 2 x 2 x 3 x 2 x 2 x 3 LCM = 4 x 3 x 2 x 2 x 3 LCM = 12 x 2 x 2 x 3 LCM = 24 x 2 x 3 LCM = 48 x 3 LCM = 144
- 46. CONTINUOUS DIVISION 36 48 Think of any number that divides evenly into both 36 and 48 (a factor of both) For this example, I’ll use 2. 2 18 24
- 47. 36 48 2 18 24 CONTINUOUS DIVISION Think of any number that divides evenly into both 18 and 24 (a factor of both) For this example, I’ll use 3. 3 Divide both numbers by 3. 6 8 2
- 48. 36 48 2 18 24 CONTINUOUS DIVISION Continue this process until there is no other number besides 1 that is a factor of both numbers. 3 3 and 4 have no common factor except 1... so you can stop dividing. 6 8 2 2 3 4
- 49. CONTINUOUS DIVISION 18 24 3 6 8 2 2 3 4 36 48 The product of the numbers outside the line is the Least Common Multiple (LCM). LCM = 2 x 3 x 2 x 3 x 4 LCM = 144
- 50. MATHEMATICS 4