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Finding LCM and GCF: Methods and Examples, Exercises of Elementary Mathematics

Kansas State University Polytechnic Campus (KSU)Elementary Mathematics

Definitions, methods, and examples for finding the least common multiple (lcm) and greatest common factor (gcf) of numbers. It covers the concept of multiples, common multiples, and the smallest common multiple (lcm). The document also explains how to find lcm using the definition and the prime number decomposition method. For gcf, it discusses the concept of divisors, common divisors, and the greatest common divisor (gcf). Methods to find gcf using the definition and the prime number decomposition method. Exercises for practice.

Typology: Exercises

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Least Common Multiple, Greatest Common Factor, Prime Factorization. Prof. L. Fern´andez
Least Common Multiple
Definitions
Recall:
•A number mis a multiple of a number aif mequals atimes a whole number.
For example, the multiples of 5 are 5,10,15,20,25,30,..., and the multiples of 3 are 3,6,9,12,15,18,... A number
has infinitely many multiples.
•Acommon multiple of two or more numbers is a number that is a multiple of all of them.
For example, 15 is a common multiple of 5 and 3 because it is a multiple of 5 and also a multiple of 3.
Two numbers have infinitely many common multiples. For example, the common multiples of 5 and 3 are 15,30,45,60,....
•The Least Common Multiple (LCM) of two or more numbers is the smallest common multiple of the given
numbers.
For example, the LCM of 3 and 5 is 15.
How to find the LCM - method 1: just use the definition.
For each number, write down the list of its multiples and find the first number that appears in all the lists (eventually,
one common multiple will appear). Drawback: This method is easy, but the lists can be very long!
Example: Find the LCM of 8 and 6:
Multiples of 8: 8,16,24,32,40,48,...
Multiples of 6: 6,12,18,24,30,36,42,48,...
The first number that appears in both lists is 24. Therefore the LCM of 8 and 6 is 24.
(Note that 48 is also a common multiple, but we are only looking for the smallest.)
Use the method above to find the LCM of the following numbers:
1. LCM of 6 and 9. 2. LCM of 4 and 6. 3. LCM of 12 and 8. 4. LCM of 6, 4 and 9.
5. LCM of 20 and 25. 6. LCM of 3 and 9. 7. LCM of 8 and 4. 8. LCM of 10, 15 and 4.
How to find the LCM - method 2: use the prime number decomposition of the numbers.
This method works well in all cases, and does not take long:
(a) Write the prime number factorization of all the numbers (review of this is done below).
(b) From all the factorizations, make a list of each factor raised to the highest exponent it has in the factorizations.
(c) Multiply the numbers of the list you got in (b).
Example: Find the LCM of 8 and 6.
(a) 8 = 23. 6 = 2 ·3.
(b) Factors: 2 with exponent 3 (2 appears in both factorizations, and the highest exponent is 3),
and 3 with exponent 1. That gives 23= 8 and 31= 3.
(c) Multiply: 8 ·3 = 24. So 24 is the LCM of 8 and 6.
Use this method to find the LCM of the following numbers:
9. LCM of 6 and 9. 10. LCM of 4 and 6. 11. LCM of 12 and 8. 12. LCM of 6, 4 and 9.
13. LCM of 20 and 25. 14. LCM of 3 and 9. 15. LCM of 8 and 4. 16. LCM of 10, 15 and 4.
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Least Common Multiple, Greatest Common Factor, Prime Factorization. Prof. L. Fern´andez

Least Common Multiple Definitions Recall:

  • A number m is a multiple of a number a if m equals a times a whole number. For example, the multiples of 5 are 5, 10 , 15 , 20 , 25 , 30 ,.. ., and the multiples of 3 are 3, 6 , 9 , 12 , 15 , 18 ,... A number has infinitely many multiples.
  • A common multiple of two or more numbers is a number that is a multiple of all of them. For example, 15 is a common multiple of 5 and 3 because it is a multiple of 5 and also a multiple of 3. Two numbers have infinitely many common multiples. For example, the common multiples of 5 and 3 are 15, 30 , 45 , 60 ,.. ..
  • The Least Common Multiple (LCM) of two or more numbers is the smallest common multiple of the given numbers. For example, the LCM of 3 and 5 is 15.

How to find the LCM - method 1: just use the definition. For each number, write down the list of its multiples and find the first number that appears in all the lists (eventually, one common multiple will appear). Drawback: This method is easy, but the lists can be very long! Example: Find the LCM of 8 and 6: Multiples of 8: 8, 16 , 24 , 32 , 40 , 48 ,... Multiples of 6: 6, 12 , 18 , 24 , 30 , 36 , 42 , 48 ,... The first number that appears in both lists is 24. Therefore the LCM of 8 and 6 is 24. (Note that 48 is also a common multiple, but we are only looking for the smallest.) Use the method above to find the LCM of the following numbers:

  1. LCM of 6 and 9. 2. LCM of 4 and 6. 3. LCM of 12 and 8. 4. LCM of 6, 4 and 9.
  2. LCM of 20 and 25. 6. LCM of 3 and 9. 7. LCM of 8 and 4. 8. LCM of 10, 15 and 4.

How to find the LCM - method 2: use the prime number decomposition of the numbers. This method works well in all cases, and does not take long: (a) Write the prime number factorization of all the numbers (review of this is done below). (b) From all the factorizations, make a list of each factor raised to the highest exponent it has in the factorizations. (c) Multiply the numbers of the list you got in (b). Example: Find the LCM of 8 and 6. (a) 8 = 2^3. 6 = 2 · 3. (b) Factors: 2 with exponent 3 (2 appears in both factorizations, and the highest exponent is 3), and 3 with exponent 1. That gives 2^3 = 8 and 3^1 = 3. (c) Multiply: 8 · 3 = 24. So 24 is the LCM of 8 and 6. Use this method to find the LCM of the following numbers:

  1. LCM of 6 and 9. 10. LCM of 4 and 6. 11. LCM of 12 and 8. 12. LCM of 6, 4 and 9.
  2. LCM of 20 and 25. 14. LCM of 3 and 9. 15. LCM of 8 and 4. 16. LCM of 10, 15 and 4.
  1. LCM of 18 and 15. 18. LCM of 24 and 30. 19. LCM of 10 and 25. 20. LCM of 6, 9 and 4.
  2. LCM of 30 and 35. 22. LCM of 20 and 24. 23. LCM of 11 and 55. 24. LCM of 15, 25 and 40.

Greatest Common Factor (GCF) Definitions:

  • A divisor or factor of a number is a number that divides it exactly (no remainder). For example, 4 is a divisor, or a factor, of 8 because 8 ÷ 4 = 2 exactly. However, 5 is not because the division 8 ÷ 5 has a remainder.
  • The divisors or factors of a number are all the numbers that divide it exactly. For example, the divisors or factors of 8 are 1, 2 , 4 and 8.
  • A common divisor or factor of two or more numbers is a number that is a divisor or factor of all of them. For example, the common divisors of 12 and 18 are 1, 2 , 3 and 6.
  • The Greatest Common Factor (GCF) (also called Greatest Common Divisor) of two or more numbers is the greatest number that is a common factor of the given numbers. For example, the GCF of 8, 24 and 20 is 4, because the factors of 8 are 1, 2 , 4 , 8. 1, 2 and 4 are also factors of 24 and 20, but 8 is not a factor of 20, so the greatest of all the common factors is 4.

How to find the GCF - method 1: just use the definition For each number, write all its factors, and then take the greatest number that appears in all the list. Drawback: this method can take very long if a number has many factors. Example: Find the GCF of 15 and 25. Factors of 15: 1, 3 , 5 , 15. Factors of 25: 1, 5 , 25. The common factors are 1 and 5. The GCF is therefore 5. In the following exercises, find the GCF using this method.

  1. GCF of 6 and 9. 26. GCF of 4 and 6. 27. GCF of 16 and 8. 28. GCF of 6, 12 and 9.
  2. GCF of 20 and 30. 30. GCF of 14 and 9. 31. GCF of 8 and 12. 32. GCF of 6, 12 and 18.
  1. 20 4 + 12 5 =^1260 +^2560 =^3760 (ex.) 50. 30 7 + 20 9 = + =

57. 58 +^56 − 14 = + + = 58. 53 − 14 +^56 = + + =

59. 10 7 +^16 + 15 2 = + + = 60. 72 +^13 +^25 = + + =

Prime number factorization Recall: A number is prime if it has no factors other than 1 and itself. For example, 2, 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 ,... are all prime. We could keep going for ever with the list, as there are infinitely many primes. Every whole number can be written in a unique way as a product of prime numbers. For example, 24 = 2 · 2 · 2 · 3. Instead of repeating a factor, we use exponents to make it shorter, so we would write 24 = 2^3 · 3. This is called the prime number factorization of a number. To find the prime number factorization of a number the only method is trial and error. Start checking whether 2 is a factor. If it is, divide the number by 2 and check if the result also has 2 as a factor. If it is, divide again, check again, etc. If 2 is not a factor, try 3, then 5, etc. It is often written in a tree. For example

24 2 12 2 6 2 3 Gives 24 = 2^3 · 3.

Gives 36 = 2^2 · 32.

Gives 70 = 2 · 5 · 7.

Gives 300 = 2^2 · 3 · 52.

Note that all the numbers at the end of the tree have to be prime. Otherwise, they can branch further.

In the following exercises, find the prime number factorization of the given numbers (just a few here – there are many more in the GCF and LCD worksheets).

  1. 16 62. 60 63. 54
  2. 15 65. 142 66. 320