Algorithms for Divisibility

In mathematics, we use and follow algorithms all the time. We follow steps and processes to calculate and solve problems. A great example to look at is using algorithms to determine whether a number is divisible by other numbers. We call these divisibility tests.

Let's begin with a simple divisibility test.

Divisibility Algorithm for testing the divisibility of 10

How do you know if a number is divisible by $$10?

That's easy, right? You just check to see if it ends in a zero.

That's an algorithm right there. You just specified a test and a check to sort numbers for those that are divisible by $$10 and for those that are not. It's a type of "search and sort" algorithm.

Example

Examine the following list of number and use the divisibility test for $$10 to determine which numbers are divisible by $$10

$$24  $$350 $$752 $$1060  $$2001  $$10 $$50 $$30021  $$52700  

Using our simple algorithm, we search each number for a zero on the end and then we sort those that do have a zero on the end from those that don't. So we get the following numbers as being divisible by 10.

$$350 $$1060  $$10  $$50 $$52700

Divisibility Algorithms for 2 and 5

To test whether numbers are divisible by 2 or 5, we have similar very simple algorithms to follow.

Try those two tests for the following list of numbers.

What did you notice about $$150?

Divisibility Algorithm for 4

To test whether a number is divisible by $$4 requires a little more work. The algorithm is as follows:

“If the last two digits are divisible by $$4, then the whole number is divisible by $$4.”

example

Show the use of the divisibility by $$4 algorithm to test whether $$52636 is divisible by $$4.

To use the algorithm, we take the last two digits and determine whether they are a multiple of $$4 (can be divided evenly by $$4).

$$364​=9

So this tells us that $$52636 can be divided by $$4.

Divisibility Algorithm for 8

To test whether a number is divisible by $$8 we use the following algorithm:

“If the last three digits are divisible by $$8, then the whole number is divisible by $$8.”

Try this algorithm for yourself with the number $$12360

You might like to use short division to see whether $$360 can be divided evenly by $$8. You can do a final check by dividing $$12360 by $$8 on your calculator.

Divisibility Algorithm for 3

To test whether a number is divisible by $$3, we use the following algorithm:

Step 1: Find the sum of the digits of the number.

Step 2: If the sum is a multiple of $$3, then the number itself is a multiple of $$3.

example

Show the use of the divisibility by $$3 algorithm to test whether $$40356 is divisible by $$3.

Step 1: $$4+0+3+5+6=18

Step 2: $$183​=6

So yes, the number $$40356 is divisible by $$3.

If that sum, $$18, had been too large and we still weren't sure whether it was divisible by $$3, we could add the digits again, giving us $$9, and look at whether that was divisible by $$3, which of course it is!

Divisibility Algorithm for 9

To test whether a number is divisible by $$9, we use the following algorithm.

Step 1: Add all the digits together

Step 2: If the sum is divisible by $$9 then the number is also divisible by $$9.

This is very similar to the algorithm for $$3, so it's your turn to have a go.

Use the algorithm for the divisibility by $$9 to determine whether $$1111707 is divisible by $$9. Check to see if you were writing by using your calculator.

Divisibility Algorithm for 6

To test whether a number is divisible by $$6, we use the following algorithm:

Step 1: Test whether the number is divisible by$$2

Step 2: Test whether the number is divisible by $$3

Step 3: If the number is divisible by both$$2 and$$3, then the number is divisible by $$6.

Again, since we've been through what's required for Step 1 and Step 2, you can have a go and test whether the following two numbers are divisible by 6. You can then check your answer on your calculator.

Is $$271926 divisible by $$6?

Is $$487421 divisible by $$6?

Divisibility Algorithm for 7

To test whether a number is divisible by $$7, we apply the following algorithm:

Step 1: Remove the units digit from the number to form two separate numbers.

Step 2: Subtract the units value from the remaining digits twice.

Step 3 (Optional): Repeat steps 1 and 2 until the number is small enough.

Step 4: It the final answer is divisible by $$7, then the whole number is divisible by $$7.

example

Show the use of the divisibility by $$7 algorithm to determine whether $$38003 is divisible by $$7.

Let's set out the required steps below.

$$3800−2×3=3794

$$379−2×4=371

$$37−2×1=35

$$35 is divisible by $$7 so $$38003 is also divisible by $$7

We could continue on with many other similar algorithms, but you now get a good feel for what these algorithms are like and how you use them. In the question set you'll see a few others that we haven't seen here, so just follow the steps to use them.

Worked Examples

question 1

Question 2

Question 3