In the Real World

Dancers A dance teacher is planning a dance for a show. The dancers will be arranged in rows with the same number of dancers in each row. Can a group of 12 dancers or a group of 14 dancers be arranged in more ways?

Prime Factorization5 1.

A group of 12 dancers can be arranged in more ways than a group of 14 dancers, because 12 has more factors than 14.

Finding FactorsEXAMPLE 1To answer the previous question, list all the factors of 12 and 14 by writing each number as a product of two numbers in as many ways as possible.

14: 1 14 2 7

7 2

The factors of 12 are:

The factors of 14 are:

12: 1 122 63 44 3

1, 3, 4, 6, and 12. 2,

1, 2, 7, and 14.

ANSWER

Prime Factorization5 1.

Stop when a pair of factors repeat.

Stop when a pair of factors repeat.

NOTE BOOK

Divisibility Rules for 2, 3, 5, 6, 9, and 10

A whole number is divisible by:

2 if the number is even.

5 if it ends with 5 or 0.

6 if it is even and is divisible by 3.

10 if it ends with 0.

Divisibility A number is divisible by another number if that other number is a factor of the first. Divisibility rules can help you find factors.

Prime Factorization5 1.

9 if the sum of its digits isdivisible by 9.

3 if the sum of its digits isdivisible by 3.

Using Divisibility RulesEXAMPLE 2Test 150 for divisibility by 2, 3, 5, 6, 9, and 10.

150 is even,

1 + 5 + 0 = 6,

150 ends with 0,

150 is even and divisible by 3,

150 is divisible by 2, 3, 5, 6 and 10, but not by 9.

so it is divisible by 2.

So, 150 is divisible by 3, but not divisible by 9.

so it is divisible by 5 and by 10.

so it is divisible by 6.

ANSWER

Prime Factorization5 1.

and 6 is divisible by 3, but not 9.

Prime Factorization5 1.

Primes and Composites A prime number is a whole number greater than 1 whose only factors are 1 and itself.

A composite number is a whole number greater than 1 that has factors other than 1 and itself. The number 1 is neither prime nor composite.

Classifying as Prime or Composite

Tell whether the number is prime or composite.

51 59

SOLUTION

List the factors of 51: List the factors of 59:

The number 59 is prime. Its only factors are 1 and itself.

1,

The number 51 is composite. It has factors other than 1 and itself.

EXAMPLE 3

1, 3, 17, 51. 59.

ANSWER ANSWER

Prime Factorization5 1.

SOLUTION

Prime Factorization5 1.

Prime Factorization Writing the prime factorization of a number means writing the number as the product of prime numbers.

You can use a diagram called a factor tree to write a factorization of a number.

To write the prime factorization, continue factoring until only prime factors appear in the product.

Writing a Prime Factorization

Write the prime factorization of 180.

180

1810

22 3 35

2 925

The prime factorization of 180 is 2 2 3 2 5.

Use multiplication to check your answer. CHECK

2 2 3 2 5 = 4 9 5 = 180

EXAMPLE 4

Write the original number.

Factor 180 as 10 times 18.

Factor 10 and 18.

Factor 9.

ANSWER

Prime Factorization5 1.