Divisibility Tests
Introduction
Divisibility Tests are ways of working out whether one number divides into another exactly without having to work out what the answer to the division would be. Thet are intended to be quick way of helping you to find factors of a number.
Some tests are quick and easy, such as the divisibility test for 10. We know that a number can be divided exactly by 10 if (and only if) its last digit is a '0'. For example, 2580 is divisible by ten (it ends with a '0'), but 6341 isn't (it doesn't end with a '0').
Other tests, such as the test for whether a number is divisible by 7, might take longer to use than to just do the division, but they can still help us to develop our understanding of division.
Starting Points
Here are some of the most common tests of divisibility (see the picture on the right).
Have a look at them and see which ones you already knew, and whether any of them are new to you.
Now try this activity, but do not use a calculator! Remember, you don't need to calculate the answer to the division, just say whether of not the smaller number divides into the larger one without remainder.
Also have a go at Dozens on nrich. Your task is to find the largest possible three-digit number which uses the computer's digits, and one of your own, to make a multiple of 2, 3, 4 or 6?
Further Practice
Using the diagram on the left (download here), place the numbers shown into the appropriate space in the Venn Diagram, based on whether they are exactly divisible by 2, 3, or 5.
Once you have done that, look at the numbers in the regions where the circles overlap.
What do you notice about all the numbers in the overlap between the "divisible by 2" and "divisible by 5" circles?
What about the other overlaps? Why do you think this works?
Does this always work? Is true that a number which is divisble by 2 and by 6 must also be divisible by 12? When does it work, and when doesn't it work? (Hint: think about what types of number 2, 3, and 5 are).
The answer can be found on slide 4 here
Further Challenges and Questions
Other divisibility tests
Here are some other divisibility tests, click on the number for the relevant video: 7, 11, 12 and 15 or look at the rules on the right. If you are super keen, look at these divisibility tests on wikipedia.
For more rigorous examples and proofs of why the rules on the right work, take a look at slides 5-11 here.
Problem-solving with divisibility tests
Below are different problem-solving tasks, can you use the divisibility tests to solve these problems?
Take a look at slides 12-16 here for hints or solutions.
Transum
Delightfully divisible from Transum is an interesting challenge to help you practise using the tests that you've learnt about. How close can you get to the solution?
