Exercise 24 ÷ 2 12

Exercise 35 ÷ 4 8 r 3

Exercise 40 ÷ 10 4

Exercise 30 ÷ 11 2 r 8

Exercise What are the possible remainders when you divide an integer by 5? 0, 1, 2, 3, 4

Multiple A multiple of an integer is the product of that integer and any natural number.

Multiples of 2 1 x 2 = 2 2 x 2 = 4 3 x 2 = 6 4 x 2 = 8

Multiples of 5 1 x 5 = 5 2 x 5 = 10 3 x 5 = 15 4 x 5 = 20

Multiples of –9 1 x (–9) = – 9 2 x (–9) = –18 3 x (–9) = –27 4 x (–9) = –36

Example 1 List the first four multiples of 7. 1 x 7 = 7 2 x 7 = 14

Example 1 List the first four multiples of –5. 1(–5) = –5 2(–5) = –10 3(–5) = –15 4(–5) = –20

Example List the first four multiples of 6. 6, 12, 18, 24

Example List the first four multiples of 11. 11, 22, 33, 44

Example List the first four multiples of –4. –4, –8, –12, –16

Factor A factor of an integer is any integer that divides the given integer with no remainder.

Divides The integer a divides the integer b (written a|b) if and only if b = a • k for some integer k.

Yes, 8 is a factor of 104 because 8 • 13 = 104. Example 2 Is 8 a factor of 104? 8 104 13 8 24 Yes, 8 is a factor of 104 because 8 • 13 = 104.

The factors of 18, in order of size, are 1, 2, 3, 6, 9, and 18. Example 3 List all the factors of 18. 1 x 18 = 18 2 x 9 = 18 3 x 6 = 18 The factors of 18, in order of size, are 1, 2, 3, 6, 9, and 18.

the integer ends in an even digit: 0, 2, 4, 6, or 8 Divisibility Tests the integer ends in an even digit: 0, 2, 4, 6, or 8 2

the sum of the integer’s digits is divisible by 3 Divisibility Tests the sum of the integer’s digits is divisible by 3 3

Divisibility Tests the number formed by the last two digits of the integer is divisible by 4 4

Divisibility Tests the integer ends in 0 or 5 5

the integer is divisible by both 2 and 3 Divisibility Tests the integer is divisible by both 2 and 3 6

Divisibility Tests the number formed by the last three digits of the integer is divisible by 8 8

the sum of the integer’s digits is divisible by 9 Divisibility Tests the sum of the integer’s digits is divisible by 9 9

Divisibility Tests the integer ends in 0 10

Since 72 = 49 and 82 = 64, you know that 7 < √58 < 8. Example 4 List all the factors of 58. Since 72 = 49 and 82 = 64, you know that 7 < √58 < 8. 1(58) = 58 3, 4, 5, 6, and 7 are not factors of 58. 2(29) = 58 The factors of 58 are 1, 2, 29, and 58.

Example List in order all the factors of 24. 1, 2, 3, 4, 6 ,8, 12, 24

Example List in order all the factors of 45. 1, 3, 5, 9, 15, 45

Example List in order all the factors of 63. 1, 3, 7, 9, 21, 63

Prime Number A prime number is a natural number greater than one that has exactly two positive factors: one and itself.

Composite Number Any natural number greater than one that has positive factors other than one and itself is called a composite number.

Prime or Composite Number Factors 1 1 neither 2 1, 2 prime 3 1, 3 prime

Prime or Composite Number Factors 4 1, 2, 4 composite 5 1, 5 prime 6 1, 2, 3, 6 composite

Since 17 has exactly two positive factors, 1 and 17, it is prime. Example 5 State whether 17 is prime, composite, or neither. 1 x 17 = 17 Since 17 has exactly two positive factors, 1 and 17, it is prime.

Since 57 has factors other than one and itself, it is composite. Example 6 State whether 57 is prime, composite, or neither. 1 x 57 = 57 3 x 19 = 57 Since 57 has factors other than one and itself, it is composite.

Example State whether 0 is prime, composite, or neither. neither

Example State whether 51 is prime, composite, or neither. composite

Example State whether 43 is prime, composite, or neither. prime

Example List the first eight prime numbers. 2, 3, 5, 7, 11, 13, 17, 19

Exercise Express 14 as a sum of 3 prime numbers. 2 + 5 + 7

Exercise Express 35 as a sum of 3 prime numbers. 5 + 13 + 17 3 + 13 + 19 5 + 11 + 19 7 + 11 + 17