A factor is a whole number that divides into another number exactly without leaving a remainder. When you multiply two whole numbers together to get a target number, both numbers are factors of that target number.
Example
Find the factors of 6.
solution :
We multiply whole numbers together to get 6:
1 × 6 = 6
2 × 3 = 6
Factors of 6 = 1, 2, 3, 6
A multiple is the result you get when you multiply a whole number by another whole number. Multiples are the numbers you find in a multiplication times table.
Example
Find the first five multiples of 6.
solution :
We multiply 6 by 1, 2, 3, 4, and 5:
6 × 1 = 6
6 × 2 = 12
6 × 3 = 18
6 × 4 = 24
6 × 5 = 30
Multiples of 6 = 6, 12, 18, 24, 30
The difference between factors and multiples comes down to size and quantity:
Example
Compare the factors and multiples of 8.
solution :
The factors of 8 break the number down: 1, 2, 4, 8.
The multiples of 8 scale the number up: 8, 16, 24, 32, 40...
To find all factors of a number, find all multiplication pairs that equal that number. Start from 1 and go up until the pairs repeat.
Example
Find all the factors of 12.
solution :
Step 1: 1 × 12 = 12
Step 2: 2 × 6 = 12
Step 3: 3 × 4 = 12
Step 4: 4 × 3 = 12 (Stop here because the numbers repeat)
Factors of 12 = 1, 2, 3, 4, 6, 12
To find multiples of a number, multiply that number by 1, 2, 3, 4, and so on.
Example
Find the first four multiples of 15.
solution :
Step 1: 15 × 1 = 15
Step 2: 15 × 2 = 30
Step 3: 15 × 3 = 45
Step 4: 15 × 4 = 60
Multiples of 15 = 15, 30, 45, 60
Common factors are factors that are shared by two or more numbers. To find them, list all the factors for each number and select the ones that appear in both lists.
Example
Find the common factors of 8 and 12.
solution :
Step 1: List all factors of 8.
Factors of 8 = 1, 2, 4, 8
Step 2: List all factors of 12.
Factors of 12 = 1, 2, 3, 4, 6, 12
Step 3: Identify the numbers that are in both lists.
The shared numbers are 1, 2, and 4.
Common Factors = 1, 2, 4
Common multiples are numbers that appear in the multiplication times tables of two or more numbers. To find them, list the multiples of each number and select the ones that match.
Example
Find the first two common multiples of 4 and 6.
solution :
Step 1: List the multiples of 4.
Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32...
Step 2: List the multiples of 6.
Multiples of 6 = 6, 12, 18, 24, 30, 36...
Step 3: Identify the first two matching numbers from both lists.
The matching numbers are 12 and 24.
Common Multiples = 12, 24
A number is divisible by 2 if its last digit is an even number, which means it ends in 0, 2, 4, 6, or 8.
Example
Test if 358 is divisible by 2.
solution :
Step 1: Identify the last digit of the number.
The last digit of 358 is 8.
Step 2: Check if the last digit is even.
Since 8 is an even number, the entire number is divisible by 2.
Result = 358 is divisible by 2
A number is divisible by 3 if the sum of all its digits can be divided perfectly by 3.
Example
Test if 417 is divisible by 3.
solution :
Step 1: Add all the individual digits together.
Sum = 4 + 1 + 7 = 12
Step 2: Check if that sum is divisible by 3.
12 ÷ 3 = 4
Result = 417 is divisible by 3
A number is divisible by 4 if the number formed by its last two digits can be divided perfectly by 4.
Example
Test if 5,324 is divisible by 4.
solution :
Step 1: Extract the last two digits of the number.
The last two digits form the number 24.
Step 2: Check if this two-digit number is divisible by 4.
24 ÷ 4 = 6
Result = 5,324 is divisible by 4
A number is divisible by 5 if its last digit is either 0 or 5.
Example
Test if 745 is divisible by 5.
solution :
Step 1: Identify the last digit of the number.
The last digit of 745 is 5.
Step 2: Check if the last digit matches the rule.
Since the last digit is 5, the entire number is divisible by 5.
Result = 745 is divisible by 5
A number is divisible by 6 if it passes the divisibility tests for both 2 and 3. This means it must be an even number and the sum of its digits must be divisible by 3.
Example
Test if 162 is divisible by 6.
solution :
Step 1: Check if the number is divisible by 2.
The last digit is 2, which is even, so it is divisible by 2.
Step 2: Check if the number is divisible by 3.
Sum of digits = 1 + 6 + 2 = 9
Since 9 ÷ 3 = 3, it is divisible by 3.
Step 3: Combine the results.
Since 162 is divisible by both 2 and 3, it is divisible by 6.
Result = 162 is divisible by 6
A number is divisible by 8 if the number formed by its last three digits can be divided perfectly by 8.
Example
Test if 7,168 is divisible by 8.
solution :
Step 1: Extract the last three digits of the number.
The last three digits form the number 168.
Step 2: Check if this three-digit number is divisible by 8.
168 ÷ 8 = 21
Result = 7,168 is divisible by 8
A number is divisible by 9 if the sum of all its digits can be divided perfectly by 9.
Example
Test if 2,835 is divisible by 9.
solution :
Step 1: Add all the individual digits together.
Sum = 2 + 8 + 3 + 5 = 18
Step 2: Check if that sum is divisible by 9.
18 ÷ 9 = 2
Result = 2,835 is divisible by 9
A number is divisible by 10 if its last digit is exactly 0.
Example
Test if 490 is divisible by 10.
solution :
Step 1: Identify the last digit of the number.
The last digit of 490 is 0.
Step 2: Check if the last digit matches the rule.
Since the last digit is 0, the entire number is divisible by 10.
Result = 490 is divisible by 10
A number is divisible by 11 if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is either 0 or a multiple of 11.
Example
Test if 1,353 is divisible by 11.
solution :
Step 1: Sum the digits in the odd positions (1st and 3rd digits from the left).
Sum of odd positions = 1 + 5 = 6
Step 2: Sum the digits in the even positions (2nd and 4th digits from the left).
Sum of even positions = 3 + 3 = 6
Step 3: Find the difference between the two sums.
Difference = 6 - 6 = 0
Step 4: Check if the difference matches the rule.
Since the difference is 0, the entire number is divisible by 11.
Result = 1,353 is divisible by 11
A number is divisible by 12 if it passes the divisibility tests for both 3 and 4. This means the sum of its digits must be divisible by 3, and its last two digits must form a number divisible by 4.
Example
Test if 384 is divisible by 12.
solution :
Step 1: Check if the number is divisible by 3.
Sum of digits = 3 + 8 + 4 = 15
Since 15 ÷ 3 = 5, it is divisible by 3.
Step 2: Check if the number is divisible by 4.
The last two digits form the number 84.
Since 84 ÷ 4 = 21, it is divisible by 4.
Step 3: Combine the results.
Since 384 is divisible by both 3 and 4, it is divisible by 12.
Result = 384 is divisible by 12
Every whole number is divisible by 1. To find if a number is divisible by other prime numbers like 7, 13, or 17, we simply use long division to check if it divides exactly with a remainder of zero.
Example 1
Find all the factors of 28.
solution :
Step 1: Find all multiplication pairs that equal 28, starting from 1.
1 × 28 = 28
2 × 14 = 28
3 does not divide 28 exactly.
4 × 7 = 28
5 and 6 do not divide 28 exactly.
7 × 4 = 28 (Stop here because the pairs are repeating).
Final Answer = 1, 2, 4, 7, 14, 28
Example 2
List the first six multiples of 13.
solution :
Step 1: Multiply 13 by 1, 2, 3, 4, 5, and 6 consecutively.
13 × 1 = 13
13 × 2 = 26
13 × 3 = 39
13 × 4 = 52
13 × 5 = 65
13 × 6 = 78
Final Answer = 13, 26, 39, 52, 65, 78
Example 3
Without using long division, test if the number 5,181 is divisible by 9.
solution :
Step 1: Add all the individual digits of the number together.
Sum = 5 + 1 + 8 + 1 = 15
Step 2: Check if the sum (15) can be divided perfectly by 9.
15 ÷ 9 = 1 with a remainder of 6.
Step 3: Apply the rule.
Since the sum of the digits is not divisible by 9, the entire number is not divisible by 9.
Final Answer = 5,181 is not divisible by 9
Example 4
Test if the number 9,328 is completely divisible by 11.
solution :
Step 1: Add the digits in the odd positions (1st and 3rd digits).
Sum of odd positions = 9 + 2 = 11
Step 2: Add the digits in the even positions (2nd and 4th digits).
Sum of even positions = 3 + 8 = 11
Step 3: Find the difference between the two sums.
Difference = 11 - 11 = 0
Step 4: Apply the rule.
Since the difference is 0, the number is divisible by 11.
Final Answer = 9,328 is divisible by 11
Example 5
Determine if the number 1,548 is divisible by 12.
solution :
Step 1: Test for divisibility by 3. Add all the digits together.
Sum = 1 + 5 + 4 + 8 = 18
Since 18 ÷ 3 = 6, it passes the test for 3.
Step 2: Test for divisibility by 4. Look at the last two digits.
The last two digits form the number 48.
Since 48 ÷ 4 = 12, it passes the test for 4.
Step 3: Combine the results.
Because the number is divisible by both 3 and 4, it is completely divisible by 12.
Final Answer = 1,548 is divisible by 12
Attempt the following Questions:
(1.) List all the factors of 48.
(2.) Find the first five multiples of 17.
(3.) Use the correct divisibility rule to show if 47,236 is completely divisible by 4.
(4.) Test if the number 2,583 is completely divisible by 6 without using long division.
(5.) Use the divisibility rule for 11 to check if 7,194 is completely divisible by 11.
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