Prime Numbers and Prime Factorization

PRIME NUMBERS AND PRIME FACTORIZATION

I. Divisors and Factors of a Number

Previously, you learned the names of the parts of a multiplication problem.

1. a. 6 × 2 = 12 6 and 2 are the______

b. 12 is the______

You learned the names of the parts of a division problem from your textbook.

2. a. 6 2 is the______2)12 12 is the______12 6 is the______the remainder is______

b. 1 8 is the______8)12 12 is the______-8 1 is the______4 4 is the______

The definitions of a factor of a certain whole number or the divisor of the whole number are in your textbook. The factors or divisors of a number will be natural numbers (1, 2, 3, 4, ...). REMEMBER we cannot divide by zero.

3. a. What is the smallest factor of 6?______b. What is the largest factor of 6? ______c. What is the smallest factor of 27?______d. What is the largest factor of 27? ______e. Generally speaking, the smallest factor of any number is______; the largest factor is ______.

This instructional aid was prepared by the Tallahassee Community College Learning Commons. II. Finding All Factors of a Number (Study your textbook.)

Now study these ways of writing a number as a product of two natural numbers:

10 │ 16 │ 1 × 10 = 10 │ 1 × 16 = 16 You can see that it is not 2 × 5 = 10 │ 2 × 8 = 16 necessary to try all natural 3 × │ 3 × numbers. You can stop after 4 × │ 4 × 4 = 16 you try the first factor 5 × 2 = 10 │ 5 × whose square is equal to or 6 × │ 6 × greater than the original 7 × │ 7 × number. 8 × │ 8 × 2 = 16 9 × │ 9 × 10 × 1 = 10 │ 10 × │ 11 × The factors │ 12 × of 10 are 1, 2, │ 13 × 5, 10 │ 14 × │ 15 × │ 16 × 1 = 16 The factors of 16 are 1, 2, 4, 8 and 16.

II. 1. Tell how you would find all the factors of 20.

a. List all______numbers, stopping with the first number whose square is ______to or______than 20. b. ______by each of these numbers. c. The factors of 20 are both the divisors and quotients of the divisions which have reminders of______.

2. Find all divisors of each number.

a. 36 b. 42 c. 37 d. 20

3. Your work will be easier if you learn the divisibility rules in your textbook.

a. A number is divisible by 2 if______b. A number is divisible by 3 if______c. A number is divisible by 5 if______

4. Look at the factors of 37.

a. There are exactly 2 different factors of 37; what are they?______, ______. b. 37 is a ______number because its only factors are 1 and the number itself. 5. Look at the factors of 36. This instructional aid was prepared by the Tallahassee Community College Learning Commons. a. Are 1 and 36 its only factors?______b. What kind of number is 36?______

6. Find all prime numbers less than 20.______

III. The Prime Factorization of a Number.

It is very important that you know what a prime factorization of a number is. A factorization of a number is a multiplication problem whose product is the number.

a. 1 × 8 All of these are factorizations of 8 because b. 2 × 4 the product is 8 in each case. c. 2 × 2 × 2

NOTICE all of the factors in "c" are prime numbers. We say 2 x 2 x 2 is the prime factorization of 8.

To find the prime factorization of a number, divide by prime numbers only. (REMAINDERS MUST BE ZERO)! Continue until the quotient is a prime number.

The prime factorization is the expression which shows all of the prime divisors and the final prime quotient as factors.

Study the prime factorization of 36. (Compare this with 2a in Part II above.)

3 3) 9 2)18 2)36

2 × 2 × 3 × 3 is the prime factorization of 36.

Students sometimes miss test questions because they aren't sure what is meant. Don't let this happen to you. Earlier, you learned to find all factors of a number and you learned to find the prime factorization of a number.

In 1 and 2 below, you answer the question. In 3 and 4, you write the question so that the answer is correct.

1. Find the prime factorization of 70.

(Your answer must be a multiplication problem in which all factors are primes.)

2. Find all the factors of 70.

(This answer is a list of all the numbers that divide evenly into 70. The numbers are separated with commas.)

This instructional aid was prepared by the Tallahassee Community College Learning Commons. 3 and 4. Study. Then tell what was found!

3. Find______of 45.

WORK 1 × 45 ANSWER: 1, 3, 5, 9, 15, 45 2 3 × 15 4 5 × 9 6 7

4. Find______of 45.

WORK 5 ANSWER: 3)15 3 × 3 × 5 3)45

5. Find the prime factorization of each composite number. If the number is prime, write "PRIME". a. 50 b. 48 c. 27 d. 29

ANSWERS:

I. 1. a. 6, 2 are factors b. 12 is the product 2. a. 2 is divisor 12 is dividend 6 is quotient 0 is remainder b. 8 is divisor 12 is dividend 1 is quotient 4 is remainder 3. a. 1 b. 6 c. 1 d. 27 e. smallest factor is 1; largest is the number itself.

II. 1. a. natural, equal, greater b. divide c. zero 2. a. 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 b. 42: 1, 2, 3, 6, 7, 14, 21, 42 c. 37: 1, 37 d. 20: 1, 2, 4, 5, 10, 20 3. a. last digit is 0, 2, 4, 6, or 8 b. sum of digits is divisible by 3 c. last digit is 0 or 5 4. a. 1, 37 b. prime

This instructional aid was prepared by the Tallahassee Community College Learning Commons. Answers (continued)

II. (continued)

5. a. No

b. composite

6. 2, 3, 5, 7, 11, 13, 17, 19 (One is not prime. It has only one factor.)

III. 1. 2 × 5 × 7 2. 1, 2, 5, 7, 10, 14, 35, 70

3. All the factors

4. Prime factorization

5. a. 2 × 5 × 5

b. 2 × 2 × 2 × 2 × 3

c. 3 × 3 × 3

d. prime

This instructional aid was prepared by the Tallahassee Community College Learning Commons.