Find Factors and Multiples of Whole Numbers - Grade 6

Factors

Division and multiplication are related operations.

• The division \(6 \div 3 = 2\) (with remainder equal to 0) may be written as the multiplication: \(6 = 2 \times 3\)
• The division \(15 \div 3 = 5\) (with remainder equal to 0) may be written as the multiplication: \(15 = 5 \times 3\)
• The multiplication \(10 = 2 \times 5\) may be written as the division: \(10 \div 5 = 2\) (with remainder equal to 0)
• The multiplication \(18 = 6 \times 3\) may be written as the division: \(18 \div 3 = 6\) (with remainder equal to 0)

Examples

• 1) \(6 = 1 \times 6\) ; \(6 = 2 \times 3\). 1, 2, 3 and 6 are called factors of 6.
• 2) \(12 = 12 \times 1 = 3 \times 4 = 6 \times 2\). 1, 2, 3, 4, 6 and 12 are called factors of 12.
• 3) \(20 = 1 \times 20 = 2 \times 10 = 2 \times 2 \times 5 = 4 \times 5\). 1, 2, 3, 4, 5, 10 and 20 are called factors of 20.

Example: 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.

• \(30 \div 1 = 30\) remainder 0 or \(30 = 30 \times 1\). Hence 1 and 30 are factors.
• \(30 \div 2 = 15\) remainder 0 or \(30 = 15 \times 2\). Hence both 2 and 15 are factors.
• \(30 \div 3 = 10\) remainder 0 or \(30 = 10 \times 3\). Hence both 3 and 10 are factors.
• \(30 \div 4 = 7\) remainder 2. 4 is not a factor.
• \(30 \div 5 = 6\) remainder 0 or \(30 = 6 \times 5\). Hence both 5 and 6 are factors.

We stop here because all factors greater than 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:

• \(5 \times 1 = 5\)
• \(5 \times 2 = 10\)
• \(5 \times 3 = 15\)
• \(5 \times 4 = 20\)

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. True or False Statements

   1. FALSE: \(4 \div 2 = 2\) with remainder 0.
   2. TRUE: \(9 = 3 \times 3\).
   3. TRUE: \(15 = 5 \times 3\).
   4. FALSE: There is no whole number \(N\) such that \(7 = 3 \times N\).
   5. TRUE: \(10 \div 10 = 1\) with remainder equal to 0.
   6. TRUE: \(16 = 16 \times 1\).
   7. TRUE: \(N = N \times 1\).
   8. TRUE: \(N = N \times 1\), meaning it has at least two factors: 1 and \(N\) (itself).

2. Finding All Factors

   1. Factors of 2:

      \(2 \div 1 = 2\) with remainder 0, or \(2 = 2 \times 1\). Hence the factors of 2 are 1 and 2.

   2. Factors of 6:

      \(6 \div 1 = 6\) with remainder 0, or \(6 = 6 \times 1\). Two factors 1 and 6.

      \(6 \div 2 = 3\) with remainder 0, or \(6 = 3 \times 2\). 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: 1, 2, 3 and 6.

   3. Factors of 10:

      \(10 \div 1 = 10\) with remainder 0, or \(10 = 10 \times 1\). Two factors 1 and 10.

      \(10 \div 2 = 5\) with remainder 0, or \(10 = 5 \times 2\). Two factors 2 and 5.

      \(10 \div 3 = 3\) with remainder 1; 3 is not a factor of 10.

      \(10 \div 4 = 2\) with remainder 2; 4 is not a factor 10.

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

   4. Factors of 24:

      \(24 \div 1 = 24\) with remainder 0, or \(24 = 24 \times 1\). Two factors 1 and 24.

      \(24 \div 2 = 12\) with remainder 0, or \(24 = 12 \times 2\). Two factors 2 and 12.

      \(24 \div 3 = 8\) with remainder 0, or \(24 = 8 \times 3\). Two factors 3 and 8.

      \(24 \div 4 = 6\) with remainder 0, or \(24 = 6 \times 4\). Two factors 4 and 6.

      \(24 \div 5 = 4\) with remainder 4; no factors.

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

3. Factors of 120

   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.

   • \(120 \div 2 = 60\) with remainder 0, 2 is a factor of 120.
   • \(120 \div 5 = 24\) with remainder 0, 5 is a factor of 120.
   • \(120 \div 10 = 12\) with remainder 0, 10 is a factor of 120.
   • \(120 \div 11 = 10\) with remainder 10, 11 is NOT a factor of 120.
   • \(120 \div 20 = 6\) with remainder 0, 20 is a factor of 120.
   • \(120 \div 30 = 4\) with remainder 0, 30 is a factor of 120.
   • \(120 \div 50 = 2\) with remainder 20, 50 is NOT a factor of 120.

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

4. Factors of 12 and 24

   We first find all factors of 12:

   • \(12 \div 1 = 12\) with remainder 0, 1 and 12 are factors of 12.
   • \(12 \div 2 = 6\) with remainder 0, 2 and 6 are factors of 12.
   • \(12 \div 3 = 4\) with remainder 0, 3 and 4 are factors of 12.

   All factors greater than 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. First 5 Multiples

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

   1. Multiples of 2:

      \(2 \times 1 = 2\) ; \(2 \times 2 = 4\) ; \(2 \times 3 = 6\) ; \(2 \times 4 = 8\) ; \(2 \times 5 = 10\)

   2. Multiples of 11:

      \(11 \times 1 = 11\) ; \(11 \times 2 = 22\) ; \(11 \times 3 = 33\) ; \(11 \times 4 = 44\) ; \(11 \times 5 = 55\)

   3. Multiples of 25:

      \(25 \times 1 = 25\) ; \(25 \times 2 = 50\) ; \(25 \times 3 = 75\) ; \(25 \times 4 = 100\) ; \(25 \times 5 = 125\)

6. Multiples of 6 and 8

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

   1. The first 5 multiples of 6 are:

      \(6 \times 1 = 6\) ; \(6 \times 2 = 12\) ; \(6 \times 3 = 18\) ; \(6 \times 4 = 24\) ; \(6 \times 5 = 30\)

   2. The first 5 multiples of 8 are:

      \(8 \times 1 = 8\) ; \(8 \times 2 = 16\) ; \(8 \times 3 = 24\) ; \(8 \times 4 = 32\) ; \(8 \times 5 = 40\)

   24 is a common multiple to 6 and 8.

References and Links

• Middle School Math (Grades 6, 7, 8, 9) - Free Questions and Problems With Answers
• High School Math (Grades 10, 11 and 12) - Free Questions and Problems With Answers
• Primary Math (Grades 4 and 5) with Free Questions and Problems With Answers