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Adding All Nine

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!

Counting Factors

Is there an efficient way to work out how many factors a large number has?


Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.


Age 7 to 14
Challenge Level

Dozens printable sheet

You may find the article on Divisibility Tests helpful.

The problem is explained below, but you may wish to scroll to the foot of the page to watch a video of the NRICH team presenting the challenge.

Do you know a quick way to check if a number is a multiple of 2? How about 3, 4, 5..., 12..., 15..., 25...?
To start with, the interactivity below will generate two random digits.
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.

Can you describe a strategy that ensures your first 'guess' is always correct?

Clicking on the purple cog gives you a chance to change the settings.
You can vary the challenge level by changing:

  • the multiple
  • the number of digits in your target number
  • the number of digits provided by the computer.

To ensure you have some choice, make sure the number of digits provided by the computer is fewer than the number of digits in the target number.

Can you describe your strategies that ensure your first 'guess' is always correct for a variety of settings?

Here is a video of the NRICH team presenting the challenge. You could just watch the start to check that you understand the problem, or you may like to pause the video and work on the task at various points.

Something else to think about:
What is the largest possible five-digit number divisible by 12 that you can make from the digits 1, 3, 4, 5 and one more digit? 

Once you've had a chance to think about this, click below to check.

Many people think the largest possible five-digit number is 53184, but there are larger ones...

Here is a selection of follow-up problems you may now like to try:
Factors and Multiples Game
Take Three from Five
American Billions

Click here for a poster of this problem

We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of this resource.