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