Find Factors and Multiples of Whole Numbers - Grade 6

Examples and questions on how to find factors and multiples of whole numbers , with detailed solutions and explanations, for grade 6 students are presented.

Factors.

Division and multiplication are related operations.

The division 6 ÷ 3 = 2 (with remainder equal to 0) may be written as the multiplication: 6 = 2 × 3

The division 15 ÷ 3 = 5 (with remainder equal to 0) may be written as the multiplication: 15 = 5 × 3

The multiplication: 10 = 2 × 5 may be written as the division: 10 ÷ 5 = 2 (with remainder equal to 0)

The multiplication: 18 = 6 × 3 may be written as the division: 18 ÷ 3 = 6 (with remainder equal to 0)

To factor a whole number is to write it as a product of two or more whole numbers.

Examples

1) 6 = 1 × 6 ; 6 = 2 × 3     1, 2, 3 and 6 are called factors of 6.

2) 12 = 12 ×1 = 3 × 4 = 6 ×2     1, 2, 3, 4, 6 and 12 are called factors of 12.

3) 20 = 1 × 20 = 2 × 10 = 2 × 2 × 5 = 4 × 5     1, 2, 3, 4, 5, 10 and 20 are called factors of 20.

How to find all factors of a whole number?.

Examples

Find all factors of 30

Divide 30 by all numbers starting from 1 and select the division that gives a remainder equal to 0 then write the factors.

30 ÷ 1 = 30 remainder 0 or 30 = 30 × 1. Hence 1 and 30 are factors

30 ÷ 2 = 15 remainder 0 or 30 = 15 × 2 . Hence both 2 and 15 are factors

30 ÷ 3 = 10 remainder 0 or 30 = 10 × 3 . Hence both 3 and 10 are factors

30 ÷ 4 = 7 remainder 2 , 4 is not a factor

30 ÷ 5 = 6 remainder 0 or 30 = 6 × 5 . Hence both 5 and 6 area factors.

We stop here because all factors greater that 5 have already been found.

The factors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30

Multiples

Take a whole number and multiply it by 1, 2, 3, 4, ... to obtain the multiples as follows:

5 × 1 = 5

5 × 2 = 10

5 × 3 = 15

5 × 4 = 20

The results of the multiplication of 5 by 1, 2, 3, 4, ... which are 5, 10, 15, 20, ... are called the multiples of 5.

    Answer the Following Questions

    1. Which of the following statements is TRUE and which is FALSE? Explain your answer.
      1. The division of 4 by 2 is equal to 2 with remainder not equal to zero.
      2. 3 is a factor of 9.
      3. 5 and 3 are factors of 15
      4. 3 is a factor of 7.
      5. The division of 10 by 10 gives a remainder equal to 0.
      6. 16 is a factor of 16.
      7. 1 is a factor to all whole numbers.
      8. Any whole number greater than 1 has at least two factors.

    2. Find all factors to the whole numbers below.
      1. 2
      2. 6
      3. 10
      4. 24

    3. Which of the following numbers is a factor of 120?
      1. 2
      2. 5
      3. 10
      4. 11
      5. 20
      6. 30
      7. 50

    4. Find and order from smallest to largest all factors of 12 and all factors of 24. Find all factors that are common (the same) to 12 and 24.

    5. Find the first 5 multiples of each of the numbers below.
      1. 2
      2. 11
      3. 25

    6. Find the first 5 multiples of 6 and 8. Find a multiple that is common (the same) to both 6 and 8.

    Solutions to the Above Problems

    1. Solution
      1. The division of 4 by 2 is equal to 2 with remainder not equal to zero.

        FALSE 4 ÷ 2 = 2 with remainder 0

      2. 3 is a factor of 9.

        TRUE 9 = 3 × 3

      3. 5 and 3 are factors of 15

        TRUE 15 = 5 × 3

      4. 3 is a factor of 7.

        FALSE There is no whole number N such that 7 = 3 × N

      5. The division of 10 by 10 gives a remainder equal to 0.

        TRUE 10 ÷ 10 = 1 with remainder equal to 0.

      6. 16 is a factor of 16.

        TRUE 16 = 16 × 1

      7. 1 is a factor to all whole numbers.

        TRUE N = N × 1

      8. Any whole number greater than 1 has at least two factors.

        TRUE N = N × 1 , at least two factors 1 and N (itself)

    2. Solution
      1. 2

        2 ÷ 1 = 2 with remainder 0 or 2 = 2 × 1 . Hence the factors of 2 are 1 and 2.

      2. 6

        6 ÷ 1 = 6 with remainder 0 or 6 = 6 × 1. Two factors 1 and 6.

        6 ÷ 2 = 3 with remainder 0 or 6 = 3 × 2. Two factors 2 and 3.

        All factors greater than 2 already found, we stop the division process and write the list of factors of 6 are 1, 2, 3 and 6.

      3. 10

        10 ÷ 1 = 10 with remainder 0 or 10 = 10 × 1. Two factors 1 and 10.

        10 ÷ 2 = 5 with remainder 0 or 10 = 5 × 2. Two factors 2 and 5.

        10 ÷ 3 = 3 with remainder 1; 3 is not a factor of 10.

        10 ÷ 4 = 2 with remainder 2; 4 is not a factor 10.

        All factors greater that 4 already found, we stop the division process and write the list of factors 10 are 1, 2, 5 and 10.

      4. 24

        24 ÷ 1 = 24 with remainder 0 or 24 = 24 × 1. Two factors 1 and 24.

        24 ÷ 2 = 12 with remainder 0 or 24 = 12 × 2. Two factors 2 and 12.

        24 ÷ 3 = 8 with remainder 0 ; or 24 = 8 × 3. Two factors 3 and 8.

        24 ÷ 4 = 6 with remainder 0; or 24 = 6 × 4. Two factors 4 and 6.

        24 ÷ 5 = 4 with remainder 4; no factors.

        All factors greater that 5 already found, we stop the division process and write the list of factors 24 are 1, 2, 3, 4, 6, 8, 12 and 24.

    3. Solution
      We divide 120 by each of the numbers in the list. If the remainder is equal to zero the divisor (number in the list) is a factor of 120.

      120 ÷ 2 = 60 with remainder 0 , 2 is a factor of 120

      120 ÷ 5 = 24 with remainder 0 , 5 is a factor of 120

      120 ÷ 10 = 12 with remainder 0 , 10 is a factor of 120

      120 ÷ 11 = 10 with remainder 10 , 11 is NOT a factor of 120

      120 ÷ 20 = 6 with remainder 0 , 20 is a factor of 120

      120 ÷ 30 = 4 with remainder 0 , 30 is a factor of 120

      120 ÷ 50 = 2 with remainder 20 , 50 is NOT a factor of 120

      In the given list all are factors of 120 except 11 and 50.

    4. Solution

      We first find all factors of 12

      12 ÷ 1 = 12 with remainder 0 , 1 and 12 are factors of 12

      12 ÷ 2 = 6 with remainder 0 , 2 and 6 are factors of 12

      12 ÷ 3 = 4 with remainder 0 , 3 and 4 are factors of 12

      All factors greater that 3 already found, we stop the division process and write the list of factors of 12: 1, 2, 3, 4, 6 and 12.

      The factors of 24 were found in solution to 2) part d) and are given by

      1, 2, 3, 4, 6, 8, 12 and 24

      Common factors to 12 and 24 are: 1, 2, 3, 4, 6 and 12.

    5. Solution

      The first 5 mutliples mutliples of a number are obtained by multiplying that number by 1, 2, 3, 4 and 5

      1. The first 5 multiples of 2 are:

        2 × 1 = 2

        2 × 2 = 4

        2 × 3 = 6

        2 × 4 = 8

        2 × 5 = 10

      2. The first 5 multiples of 11 are:

        11 × 1 = 11

        11 × 2 = 22

        11 × 3 = 33

        11 × 4 = 44

        11 × 5 = 55

      3. The first 5 multiples of 25 are:

        25 × 1 = 25

        25 × 2 = 50

        25 × 3 = 75

        25 × 4 = 100

        25 × 5 = 125

    6. Solution

      The first 5 mutliples mutliples of 6 and 8 number are obtained by multiplying 6 and 8 by 1, 2, 3, 4 and 5

      1. The first 5 multiples of 6 are:

        6 × 1 = 6

        6 × 2 = 12

        6 × 3 = 18

        6 × 4 = 24

        6 × 5 = 30

      2. The first 5 multiples of 8 are:

        8 × 1 = 11

        8 × 2 = 16

        8 × 3 = 24

        8 × 4 = 32

        8 × 5 = 40

        24 is a common multiple to 6 and 8

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