1. Lesson 2.3 Happenings at Harvest Day: Investigating Factors and Greatest Common Factors
EQ: How do we determine the greatest common factor of two or more numbers?

2. The Fundamental Theorem of Arithmetic
The Fundamental Theorem of Arithmetic states that every natural number is either prime or can be written as a unique product of primes.

3. Greatest Common Factor (GCF)
A common factor is a factor of two or more numbers.
The greatest common factor (GCF) is the greatest factor two or more numbers have in common.

4. EXAMPLE QUESTION
Pat is creating thank-you fruit baskets for the adults who will volunteer at Saturday’s Harvest Day event. She wants to make sure each basket contains an equal quantity of each type of fruit. She has 24 bananas, 12 apples, and 18 oranges.
How many different fruit baskets can she make so that each one has an equal number of each type of fruit?
1 x x x 18
2 x x x 9
3 x 8 3 x x 6
4 x 6
Common Factors are 1, 2, 3, and 6
Pat can assemble 1, 2, 3, or 6 baskets.

5. EXAMPLE QUESTION
Pat is creating thank-you fruit baskets for the adults who will volunteer at Saturday’s Harvest Day event. She wants to make sure each basket contains an equal quantity of each type of fruit. She has 24 bananas, 12 apples, and 18 oranges.
How many bananas, apples, and oranges will be in each fruit basket if Pat decides to make the greatest number of fruit baskets? Explain your reasoning.
1 x x x 18
2 x x x 9
3 x 8 3 x x 6
4 x 6
Common Factors are 1, 2, 3, and 6. GCF = 6
The greatest number of fruit baskets that Pat can make is 6.

6. EXAMPLE QUESTION
Pat is creating thank-you fruit baskets for the adults who will volunteer at Saturday’s Harvest Day event. She wants to make sure each basket contains an equal quantity of each type of fruit. She has 24 bananas, 12 apples, and 18 oranges.
How many bananas, apples, and oranges will be in each fruit basket if Pat decides to make the greatest number of fruit baskets? Explain your reasoning.
1 x x x 18
2 x x x 9
3 x 8 3 x x 6
4 x 6
If there are 6 fruit baskets, then there will be 4 bananas, 2 apples, and 3 oranges in each basket. I know that Pat needs to make sure each basket has the same amount of each type of fruit. So, she can calculate the greatest number of baskets possible (to have the same amount of each type of fruit), and then she can divide each quantity of fruit by the number of baskets.

7. Determine the greatest common factor using prime factorization.
What is the GCF of 24 and 36?
Step 1 - Write the prime factorizations of 24 and 36 as mathematical statements
24 = 2 x 2 x 2 x 3
36 = 2 x 2 x 3 x 3
Step 2 - Write the factors common to both
The common factors of 24 and 36 are 2 x 2 x 3 or 22 x3
Step 3 - What is the greatest common factor of 24 and 36?
22 x3 = 12
The greatest common factor is 12.

8. What is the least common factor of 24 and 36?
The least common factor of 24 and 36 is 1
Explain your reasoning.
Because any set of numbers will always have a least common factor of 1.

9. YOUR TURN Determine the greatest common factor of each pair.
20 and and and 81
20 = 22 x = 22 x = 1 x 37
40 = 23 x = 24 x = 34
GCF = 22 x 5 = 20 GCF = 22 x 3 = 12 GCF = 1
The GCF of 20 and The GCF of 36 and 48 The GCF of 37 and 81
40 is is is 1.

10. Finally
Two numbers that do not have any common factors other than 1 are called relatively prime numbers.