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Find Greatest common Factor and Least common Multiple

Find the Prime Factorization of a Number 

Prime number: Prime numbers are the numbers that have only two factors, that are, one and the number itself. 

Composite number: A number that has more than two factors. 

Prime numbers:  

2, 3, 5, 7, 11, 13 ,17, 19, 

23, 29, 31, 37, 41, 43, 47, 

53, 59, 61, 67, 

71, 73, 79, 83, 89, 

97 

1-100, the total prime numbers are 25. 

Composite numbers: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 79, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100 

1-100, the composite numbers are 75. 

Prime factorization 

Whole numbers greater than 1 are either prime or composite numbers. 

A composite number can be written as a product of its prime factors, called its prime factorization. 

Example 1: 

How can you find the prime factorization of 48? 

ONE WAY 

To find the prime factorization of 48, write its factors as a product.  

  48 = 2 × 24 

= 2 × 2 × 12 

= 2 × 2 × 2 × 6  

= 2 × 2 × 2 × 2 × 3 

∴ The prime factorization of 48 is 2 × 2 × 2 × 2 × 3 or 2⁴ × 3. 

ANOTHER WAY 

A factor tree shows the prime factorization of a composite number. 

∴ The prime factorization of 48 is 2 x 2 x 2 x 2 x 3 or 2⁴ x 3 

Try It! 

Find the prime factorization of 56.  

Solution: 

Start with the least prime factor. 

The prime factorization of 56 is 2 × 2 × 2 × 7 = 23 × 7 

Convince Me! 

A number is greater than 2, and it has 2 as a factor. Is the number a prime or a composite? Explain. 

Solution:  

4 = 2 × 2 = 2²

6 = 2 × 3    

8 = 2 × 2 × 2 = 2³

Product of prime numbers is always a composite number 

Example 2: 

Keesha is putting together bags of supplies. She puts an equal number of craft sticks and an equal number of glue bottles in each bag. There are no supplies left over. What is the greatest number of bags of supplies that Keesha can make? 

Solution: 

Identify the greatest common factor (G.C.F) of 12 and 42. The G.C.F is the greatest number that is a factor of two or more numbers.  

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12 =   2    ×  2  ×   3  

12 =   2    ×  2  ×   3  

42 =    2    × 3   ×    7  .   

Multiply the common factors.  

2 × 3 = 6  

∴ The greatest common factor (G.C.F) of 12 and 42 is 6.  

∴ Keesha can make 6 bags of supplies. 

Try It!  

Keesha has 24 beads to add equally to each bag. Can she still make 6 bags and have no supplies left over? 

Solution: 

Number of craft sticks = 42 

Number of glue bottles = 12 

Number of beads = 24 

12 =   2    ×    2    ×  3 

24 =   2    ×    2    ×  2  ×  3    

42 =   2    ×    3    ×  7  

∴ G.C.F of 12, 24 and 42 is 2 × 3 = 6 

Yes, Keesha makes 6 bags and have no supplies left over. 

Example 3: 

Use the G.C.F and the distributive property to find the sum of 18 and 24. 

STEP 1 

 Find the G.C.F of 18 and 24. 

18 =   2    ×  3  ×    3                             

 24 =   2    ×  2   ×  2  ×    3    

The greatest number that is a factor of both 18 and 24 is 2 x 3. 

The G.C.F of 18 and 24 is 6. 

STEP 2 

Write each number as a product using the G.C.F as a factor. 

18 + 24 = 6 × 3 + 6 × 4  

=6(3 + 4) Apply the Distributive Property  

= 6(7)  

= 42  

The sum of 18 and 24 is 42. 

Try It!  

Use the G.C.F and the distributive property to find the sum of 12 and 36. 

Solution: 

Let’s find the G.C.F of 12 and 36 

2 =   2   ×   2   ×   3  

36 =   2   ×   2   ×   3   ×  3 

G.C.F = 2 × 2 × 3 

= 12 

Sum of 12 and 36 by G.C.F 

12 + 36 = 12 × 1 + 12 × 3  

                = 12(1 + 3) 

                = 12 × 4 

                = 48 

∴ The sum of 12 and 36 is 48      

Example 4:  

Grant is making picnic lunches. He wants to buy as many juice bottles as applesauce cups, but no more than he needs to have an equal number of each. 

How many packages of each should Grant buy? 

The least common multiple (L.C.M) is the least multiple, not including zero, common to both numbers. 

Step1:  

Write the prime factorization of each number 

           6 = 2 × 3  

           8 = 2 × 2 × 2 

Step2:     

List the greatest number of times each factor appears in either prime factorization. Multiply these factors to find the L.C.M. 

 If the same factor occurs more than once in both numbers, then multiply the factor, the maximum number of times it occurs 

The occurrence of numbers in the above example  

2: three times 

3: one time 

24 is the L.C.M of 6 and 8.                               

                               6 × 4 = 24                        8 × 3 = 24  

∴ Grant should buy 4 packages of juice and 3 packages of apple sauce. 

Try It!  

Grant also buys bottled water and juice pouches for the picnic. There are 12 bottles of water in each case and 10 juice pouches in each box. Grant wants to buy the least amount but still have as many bottles of water as juice pouches. How many of each should he buy? Explain. 

Solution: 

Given that, 

No. of bottles of water in each case = 12 

      No. of juice pouches in each box = 10  

                                 L.C.M of 12 and 10 = ? 

∴ L.C.M. of 12 and 10 = 60 

∴ Grant has to buy 60 bottled water and juice pooches for the picnic. 

Key Concepts  

The greatest common factor (G.C.F) of two numbers is the greatest number that is a factor of both numbers. 

Factors of 12: 1, 2, 3, 4, 6, 12  

Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40  

2 and 4 are common factors of 12 and 40.  

4 is the greatest common factor. 

∴ The G.C.F of 12 and 40 is 4. 

The least common multiple (L.C.M) of two numbers is the least multiple, not including zero, common to both numbers.  

Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48 …  

Multiples of 9: 9, 18, 27, 36, 45, 54…  

18 and 36 are common multiples of 6 and 9.  

18 is the least common multiple.  

∴ The L.C.M of 6 and 9 is 18. 

Practice and problem solving: 

  1. You have 50 blueberry scones and 75 cranberry scones. You want to make as many identical bags as possible. Each bag should have an equal number of blueberry scones and an equal number of cranberry scones. What is the greatest number of bags you can fill? Explain. 

Solution: 

Given that, 

No. of blueberry scones = 50 

No. of cranberry scones = 75 

G.C.F of 50 and 75 

50 =  2  ×   5   ×   5 

75 =  5  ×   5   ×  3  

G.C.F = 5 × 5 

           =25 

∴ The greatest number of bags you can fill is 25 bags 

  1. Gena has 28 trading cards, Sam has 91 trading cards, and Tiffany has 49 trading cards. Use the G.C.F and the distributive property to find the total number of trading cards Gena, Sam, and Tiffany have. 

Solution: 

Given that, 

No of trading cards with Gena= 28                              

No of trading cards with Sam=91 

No of trading cards with Tiffany = 49 

G.C.F of 28, 91 and 49 

28 =  2   ×  2  ×   7  

91 =  7   ×  13  

49 =  7   ×  7 

G.C.F = 7 

Sum of 28, 91 and 49 by G.C.F 

28 + 91 + 49 = 7 × 4 + 7 × 13 + 7 × 7 

= 7 (4 + 13 + 7) 

= 7 × 24  

                         =168 

∴ Sum of 28, 91 and 49 is 168 

  1. Rami has swimming lessons every 3 days and guitar lessons every 8 days. If he has both lessons on the first day of the month, in how many days will Rami have both lessons on the same day again? 

Solution: 

Given that, 

Swimming lessons for every 3 days 

Guitar lessons for every 8 days 

L.C.M of 3 and 8 

3 = 1 × 3 

8 = 2 × 2 × 2 

L.C.M = 1 × 3 × 2 × 2 × 2 

 = 24 

  ∴ Rami have both lessons on the same day again on 24th day 

Let’s check our knowledge 

  1. Write the prime factorization of each 33. If the number is prime, write prime?  
  1. 2 Find the G.C.F for the following pair of numbers: 100, 48  
  1. Find the L.C.M for the following pair of numbers: 8, 10  
  1. Use the G.C.F and the distributive property to find the sum of 30 and 66? 
  1. Why is the G.C.F of two prime numbers always 1? 
  1. Sarah says that you can find the L.C.M of any two whole numbers by multiplying them together. Provide a counter example to show that Sarah’s statement is incorrect? 

Answers: 

  1. Given number = 33 

                                             = 3 × 11     So, 33 is composite number. 

  1. First write the prime factorization of 100 and 48. 
    100 = 2 × 50 48 = 2 × 24 
    = 2 ×2 × 25       = 2 × 2 × 12 
    100 = 2×2×5×5       = 2 × 2 × 2 × 6 
    48 = 2×2×2×2×3 48 = 2 × 2 × 2 × 2 × 3 
    ∴ G.C.F of 100 and 48 is 2 × 2 = 4 
  1. Solution: 

First, write prime factorization of 8 and 10. 
8 =  2  ×  4                                       10 = 2 × 5 
8 = 2×2×2   10 =2× 5 

2 is the repeated factor in both, so we can write once. 

∴ L.C.M of 8 and 10 is 2 × 2 × 2 × 5 = 40 

  1. Solution: 

Given that, 

30 and 66  

30 = 2 × 3 × 5 

66 = 2 × 3 × 11 

G.C.F of 30 and 66 = 2 × 3 

                                   = 6 

Sum of 30 and 66 by G.C.F 

30 + 66 = 6 × 5 + 6 × 11 

  = 6(5 + 11) 

  = 6 × 16 

= 96 

∴ The sum of 30 and 66 is 96. 

  1. Solution: 

Let’s take two prime numbers 2 and 3. 

2 =    1    ×  2  

3 =    1    ×  3 

G.C.F = 1 

There is only one common factor for prime numbers which  is 1. 

∴ The G.C.F for prime numbers is 1. 

  1. Solution: 

Given statement is that the L.C.M of two whole numbers is their product. 

Let’s take two whole numbers 5 and 10  

5 = 1 × 5 

10 = 2 × 5 

G.C.F = 5 

The G.C.F of 5 and 10 is 5. 

∴ The given statement is incorrect. 

Exercise:

  1. Write the prime factorization of the following:
    a. 25
    b. 49
    c. 296
    d. 100
  2. Find the G.C.F of the following:
    a. 1, 200
    b. 2, 20
    c. 25, 625
    d. 36, 31
  3. Find the L.C.M. of the following:
    a. 10, 20, 24
    b. 100, 122, 3
  4. Find the sum of 21 and 27 using G.C.F. and distributive property?
  5. There are 25 cars, 75 cycles and 100 scooters in the parking yard. Find the total number of vehicles in the parking yard using G.C.F. and distributive property?
  6. A number is between 58 and 68. It has prime factors of 2,3, and 5. What is the number?

 What have we learned:

  • Understand how to write the prime factorization.
  • Understand how to find the greatest common factor of two numbers.
  • Understand how to find the least common multiple of two numbers.
  • Understand how to find the greatest common factor of two whole numbers less than or equal to 100.
  • Understand how to find the least common multiple of two whole numbers less than or equal to 12.
  • Understand how to use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor.

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