Columbus State Community College

1. Columbus State Community College Chapter 4 Section 2 Writing Fractions in Lowest Terms

2. Writing Fractions in Lowest Terms • Identify fractions written in lowest terms. • Write fractions in lowest terms using common factors. • Write a number as a product of prime factors. • Write a fraction in lowest terms using prime factorization. • Write a fraction with variables in lowest terms.

3. Note on Factors NOTE Recall that factors are numbers being multiplied to give a product. For example, 1 • 5 = 5, so 1 and 5 are factors of 5.5 • 7 = 35, so 5 and 7 are factors of 35. 5 is a factor of both 5 and 35, so 5 is a common factor of those numbers.

4. 1 5 2 11 9 8 7 34 Writing a Fraction in Lowest Terms Writing a Fraction in Lowest Terms A fraction is written in lowest terms when the numerator and denominator have no common factors other than 1. Examples are , , , and . When you work with fractions, always write the final answer in lowest terms.

5. 9 10 14 25 Identifying Fractions Written in Lowest Terms EXAMPLE 1 Identifying Fractions Written in Lowest Terms Are the following fractions in lowest terms?  The factors of 9 are 1, 3, and 9. (a)  The factors of 14 are 1, 2, 7, and 14. The numerator and denominator have no common factor other than 1, so the fraction is in lowest terms.  The factors of 10 are 1, 2, 5, and 10. (b)  The factors of 25 are 1, 5, and 25. The numerator and denominator have a common factor of 5, so the fraction is not in lowest terms.

6. 27 27 3 36 36 4 27 ÷ 9 36 ÷ 9 Using Common Factors to Write Fractions in Lowest Terms EXAMPLE 2 Using Common Factors – Lowest Terms Write each fraction in lowest terms. (a) The largest common factor of 27 and 36 is 9. Divide both numerator and denominator by 9. = =

7. 40 40 8 55 55 11 40 ÷ 5 55 ÷ 5 Using Common Factors to Write Fractions in Lowest Terms EXAMPLE 2 Using Common Factors – Lowest Terms Write each fraction in lowest terms. (b) The largest common factor of 40 and 55 is 5. Divide both numerator and denominator by 5. = =

8. 4 32 32 9 72 72 32 ÷ 8 72 ÷ 8 Using Common Factors to Write Fractions in Lowest Terms EXAMPLE 2 Using Common Factors – Lowest Terms Write each fraction in lowest terms. – (c) The largest common factor of 32 and 72 is 8. Divide both numerator and denominator by 8. – – – = = Keep the negative sign

9. 3 60 6 60 4 8 80 80 60 ÷ 10 6 ÷ 2 80 ÷ 10 8 ÷ 2 Using Common Factors to Write Fractions in Lowest Terms EXAMPLE 2 Using Common Factors – Lowest Terms Write each fraction in lowest terms. (d) Suppose we made an error and thought that 10 was the largest common factor of 60 and 80. = = = Not in lowest terms

10. Dividing by a Common Factor – Fractions in Lowest Terms Dividing by a Common Factor – Fractions in Lowest Terms Step 1Find the largest number that will divide evenly into both the numerator and denominator. This number is a common factor. Step 2 Divide both numerator and denominator by the common factor. Step 3 Check to see if the new numerator and denominator have any common factors (besides 1). If they do, repeat Steps 1 and 2. If the only common factor is 1, the fraction is in lowest terms.

11. Prime Numbers Prime Numbers A prime number is a whole number that has exactly two different factors, itself and 1. Here are the prime numbers smaller than 50. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

12. Composite Numbers Composite Numbers A number with a factor other than itself or 1 is called a composite number.

13. Zero and One CAUTION A prime number has only two different factors, itself and 1. The number 1 is not a prime number because it does not have two different factors; the only factor of 1 is 1. Also, 0 is not a prime number. Therefore, 0 and 1 are neither prime nor composite numbers.

14. Finding Prime Numbers EXAMPLE 3 Finding Prime Numbers Label each number as prime or composite or neither. Prime numbers Composite numbers Neither 0 6 7 9 12 15 17 20 29 35 42 43

15. Prime and Odd Numbers CAUTION All prime numbers are odd numbers except the number 2. Be careful though, because not all odd numbers are prime numbers. For example, 21, 25, and 27 are odd numbers but they are not prime numbers.

16. Prime Factorization Prime Factorization A prime factorization of a number is a factorization in which every factor is a prime factor. Examples Prime factorization of 60 60 = 2 • 2 • 3 • 5 Prime factorization of 126 126 = 2 • 3 • 3 • 7 All Prime Numbers

17. Methods for Finding the Prime Factorization • We will discuss two methods for finding the prime • factorization of a number. • The Division Method • The Factor Tree Method

18. Factoring Using the Division Method Let’s say we want to use the division method to find the prime factorization of 30. 2 30 30 = 2 • 3 • 5 3 15 5 5 1 You’re done!

19. Prime Factorization – the Order of Factors NOTE You may write the factors in any order because multiplication is commutative. So you could write the factorization of 30 as 5 • 3 • 2. We will show the factors from smallest to largest in our examples.

20. Factoring Using the Division Method EXAMPLE 4 Factoring Using the Division Method (a) Find the prime factorization of 84. 2 84 2 42 84 = 2 • 2 • 3• 7 3 21 7 7 You’re done! 1

21. Factoring Using the Division Method EXAMPLE 4 Factorizing Using the Division Method (b) Find the prime factorization of 150. 2 150 3 75 150 = 2 • 3 • 5• 5 5 25 5 5 You’re done! 1

22. Factoring Using the Division Method CAUTION When you’re using the division method of factoring, the last quotient is 1. Do not list 1 as a prime factor because 1 is nota prime number.

23. Factoring Using the Factor Tree Method Let’s say we want to use the factor tree method to find the prime factorization of 120. 120 120 10 12 6 20 2 5 4 3 2 3 4 5 2 2 2 2 120 = 2• 2• 2 • 3 • 5 120 = 2 • 2• 2 • 3• 5

24. Factoring Using the Factor Tree Method EXAMPLE 5 Factoring Using the Factor Tree Method (a) Find the prime factorization of 270. 270 9 30 3 3 6 5 2 3 270 = 2• 3• 3 • 3 • 5

25. Factoring Using the Factor Tree Method EXAMPLE 5 Factoring Using the Factor Tree Method (b) Find the prime factorization of 108. 108 2 54 9 6 3 3 2 3 108 = 2• 2• 3 • 3 • 3

26. Divisibility Tests NOTE Here is a reminder about the quick way to see whether a number is divisible by 2, 3, or 5; in other words, there is no remainder when you do the division. A number is divisible by 2 if the ones digit is 0, 2, 4, 6, or 8. For example, 68, 994, and 560 are all divisible by 2. A number is divisible by 3 if the sum of the digits is divisible by 3. For example, 435 is divisible by 3 because 4 + 3 + 5 = 12 and 12 is divisible by 3. A number is divisible by 5 if it has a 0 or 5 in the ones place. For example, 95, 820, and 17,225 are all divisible by 5.

27. 2 24 24 7 84 84 2 • 2 • 2 • 3 2 • 2 • 3 • 7 Using Prime Factorization – Fractions in Lowest Terms EXAMPLE 6 Using Prime Factorization to Write Fractions in Lowest Terms (a) Write in lowest terms. 24 can be written as 2 • 2 • 2 • 3  Prime factors 84 can be written as 2 • 2 • 3 • 7  Prime factors 1 1 1 = = 1 1 1

28. 6 54 54 7 63 63 2 • 3 • 3 • 3 3 • 3 • 7 Using Prime Factorization – Fractions in Lowest Terms EXAMPLE 6 Using Prime Factorization to Write Fractions in Lowest Terms (b) Write in lowest terms. 54 can be written as 2 • 3 • 3 • 3  Prime factors 63 can be written as 3 • 3 • 7  Prime factors 1 1 = = 1 1

29. 1 14 14 5 70 70 2 • 7 2 • 5 • 7 Using Prime Factorization – Fractions in Lowest Terms EXAMPLE 6 Using Prime Factorization to Write Fractions in Lowest Terms (c) Write in lowest terms. 14 can be written as 2 • 7  Prime factors 70 can be written as 2 • 5 • 7  Prime factors 1 1 = = 1 1

30. 1 1 14 5 5 70 1 1 2 • 7 = = 2 • 5 • 7 1 1 1 in the Numerator CAUTION In Example 6(c), all factors of the numerator divided out. But 1 • 1 is still 1, so the final answer is (not 5).

31. Using Prime Factorization – Fractions in Lowest Terms Using Prime Factorization to Write a Fraction in Lowest Terms Step 1 Write the prime factorization of both numerator and denominator. Step 2 Use slashes to show where you are dividing the numerator and denominator by any common factors. Step 3Multiply the remaining factors in the numerator and in the denominator.

32. 2 8 8 n 4n 4n 2 • 2 • 2 2 • 2 • n Writing Fractions with Variables in Lowest Terms EXAMPLE 7 Writing Fractions with Variables in Lowest Terms (a) Write in lowest terms. 8 can be written as 2 • 2 • 2  Prime factors 4n can be written as 2 • 2 • n  4n means 4 • n = 2 • 2 • n 1 1 = = 1 1

33. 6ab 6ab 9abc 9abc 2 2 • 3 • a • b 3c 3 • 3 • a • b • c Writing Fractions with Variables in Lowest Terms EXAMPLE 7 Writing Fractions with Variables in Lowest Terms (b) Write in lowest terms. 6ab can be written as 2 • 3 • a • b  6ab = 2 • 3 • a • b 9abc can be written as 3 • 3 • a • b • c  9abc = 3 • 3 • a • b • c 1 1 1 = = 1 1 1

34. 30 m2n5 30 m2n5 42 m3n2 42 m3n2 Writing Fractions with Variables in Lowest Terms EXAMPLE 7 Writing Fractions with Variables in Lowest Terms (c) Write in lowest terms. Alternative Method Reduce the coefficients using any method you choose. By how many? Now take care of the variables. By how many? Do you have more m’s on top or bottom? Do you have more n’s on top or bottom? n 5 3 = 7 m The 1 is optional as the exponent on the m.

35. Writing Fractions in Lowest Terms Chapter 4 Section 2 – Completed Written by John T. Wallace