At the chalkboard the teacher demonstrates the four factors of the number 6, that is, the whole number that can divide into 6 and leave no remainder. (remember, a number is always its own factor; as to 1). between 1 and 100 there are five numbers that have exactly twelve factors. find the five factors

The five numbers that have 12 factors are 96 , 90 , 84 , 72 , 60

Step-by-step explanation:

* Lets explain how to solve the problems

- The factors of 6 are 1 , 2 , 3 , 6 ⇒ 4 factors

- The factors of 12 are 1 , 2 , 3 , 4 , 6 , 12 ⇒ 6 factors

- We need a number with 12 factors twice the number of factors of 12

∴ We look for a multiple of 12 less than 100

- Lets start with 96 the greatest multiple of 12 and less than 100

∵ The factors of 96 are:

# 1 × 96

# 2 × 48

# 3 × 32

# 4 × 24

# 6 × 16

# 8 × 12

∴ 96 has 12 factors

∵ 84 is a multiple of 12

# 1 × 82

# 2 × 41

# 3 × 28

# 4 × 21

# 6 × 14

# 7 × 12

∴ 84 has 12 factors

∵ 72 is a multiple of 12

# 1 × 72

# 2 × 36

# 3 × 24

# 4 × 18

# 6 × 12

# 8 × 9

∴ 72 has 12 factors

∵ 60 is a multiple of 12

# 1 × 60

# 2 × 30

# 3 × 20

# 4 × 15

# 5 × 12

# 6 × 10

∴ 60 has 12 factors

- Now we have four numbers any multiple of 12 less than 60 has

 number of factors less than 12

- So lets look for a multiple of 6 and less than 100

- The greatest multiple of 6 and less than 100 is 96 but we took it

  before so lets take 90

∵ 90 is a multiple of 6

# 1 × 90

# 2 × 45

# 3 × 30

# 5 × 18

# 6 × 15

# 9 × 10

∴ 90 has 12 factors

* The five numbers that have 12 factors are 96 , 90 , 84 , 72 , 60