Divisibility rules: word problems

key notes :

Understanding Divisibility Rules

  • Divisibility by 2: A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8.
  • Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
  • Divisibility by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
  • Divisibility by 5: A number is divisible by 5 if it ends in 0 or 5.
  • Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3.
  • Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
  • Divisibility by 10: A number is divisible by 10 if it ends in 0.

Applying Divisibility Rules to Word Problems

  • Identify Key Information: Look for numbers and divisibility criteria given in the problem.
  • Apply Divisibility Rules: Use the rules to determine if the number can be divided without a remainder.
  • Solve the Problem: Use the results from applying divisibility rules to answer the question asked.

Learn with an example

Tessa wants to arrange 132 roses into vases. She wants to put the same number of roses in each vase without any roses left over. How many vases could Tessa use?

  • 3
  • 5
  • 10
  • 9

Which number evenly divides 132? Try each number.

First try the “divisible by 3” rule on 132.

Find the sum of the digits:

1 + 3 + 2 = 6

6 is divisible by 3. 
The rule says that 132 is divisible by 3.

Next try the “divisible by 5” rule on 132.

Look at the ones digit:

132

The ones digit is 2. 
The rule says that 132 is not divisible by 5.

Next try the “divisible by 10” rule on 132.

Look at the ones digit:

132

The ones digit is 2. 
The rule says that 132 is not divisible by 10.

Finally, try the “divisible by 9” rule on 132.

Find the sum of the digits:

1 + 3 + 2 = 6

6 is not divisible by 9. 
The rule says that 132 is not divisible by 9.

132 is divisible by 3. Tessa could use 3 vases.

A biscuit factory made 1,092 oatmeal biscuits. After arranging the biscuits into packages, the factory did not have any biscuits left over. How many biscuits could have gone in each package?

  • 6
  • 9
  • 5
  • 10

Which number evenly divides 1,092? Try each number.

First try the “divisible by 6” rule on 1,092.

First check if 1,092 is divisible by 2.

Look at the ones digit:

1,092

The ones digit is 2. 

So, the number is divisible by 2.

Now check if 1,092 is divisible by 3.

Find the sum of the digits:

1 + 0 + 9 + 2 = 12

12 is divisible by 3. So, the number is divisible by 3.
The rule says that 1,092 is divisible by 6.

Next try the “divisible by 9” rule on 1,092.

Find the sum of the digits:

1 + 0 + 9 + 2 = 12

12 is not divisible by 9. 
The rule says that 1,092 is not divisible by 9.

Next try the “divisible by 5” rule on 1,092.

Look at the ones digit:

1,092

The ones digit is 2. 
The rule says that 1,092 is not divisible by 5.

Finally, try the “divisible by 10” rule on 1,092.

Look at the ones digit:

1,092

The ones digit is 2. 
The rule says that 1,092 is not divisible by 10.

1,092 is divisible by 6. Each package could have 6 biscuits.

Let’s practice!