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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

Why do this problem?

This problem offers a twist on the usual way of assessing students' knowledge of divisibility tests. Rather than asking students to check whether a number is divisible by 2, 3, 4, 5... students have to puzzle over the choices available as they are challenged to find the largest number that meets the necessary criteria. This low threshold high ceiling task has an accessible starting point, but then offers increasing levels of challenge as students can opt to work with multiples of larger and larger numbers..., and they may sometimes find that it is impossible to meet the criteria!


Possible approach

You may find the article on Divisibility Tests helpful. It offers clear explanations of the various divisibility rules and why they work.
This problem featured in an NRICH webinar in June 2020.
You can introduce the problem by showing the default setting of the interactivity with multiples of 2. Students could write the answer to the computer-generated problem on individual student whiteboards, or they could discuss with their partner before offering their suggestion. If the computer does not agree with the class's suggestion, take some time to discuss why. Repeat this a few times and challenge the class to develop a strategy for ALWAYS finding the highest even number on their first attempt.
If you do not have access to the interactivity, you can model what the computer does using a pack of cards. 
Depending on the experience of your students, you'll need to judge which multiples to move on to next. With younger students, it may be appropriate to move on to multiples of 5, then 10, then 20 (which may introduce some impossible situations). With older students, it may be more appropriate to move on to multiples of 20, then 3, then 4, then 6.
Along the way, you'll need to bring the class together to discuss different divisibility rules and why they work. 
Ideally, pairs of students can work together at a tablet or computer to develop their strategy. Alternatively, each pair could be given a set of digit cards and select two at random to model the interactivity. Here are a couple of ways in which students could work together:
  • Collaboratively - students work on a particular multiple until they have entered five correct solutions in a row, on their first attempt before moving on to another multiple;
  • Competitively - each student in the pair works independently to find the largest number and a point is awarded if only one of them has the correct solution. (Students who know their divisibility rules will usually draw!)
Encourage students to change the settings of the interactivity themselves. Here are some mathematically interesting possibilities:
Target number: 3 digits
Numbers provided: 2 digits
Multiple of: 15 (sometimes impossible)
Target number: 3 digits
Numbers provided: 2 digits
Multiple of: 12 (sometimes impossible)
Target number: 4 digits
Numbers provided: 2 digits
Multiple of: 20
Target number: 4 digits
Numbers provided: 2 digits
Multiple of: 200 (sometimes impossible)
While the class is working, circulate round the room looking out for pairs who are becoming expert at a particular setting. When you bring the class together, you could select pairs of students to explain the strategies they have developed.
If you or your students have kept a record of impossible situations, you might want to display them on the board and then discuss why they do not have a solution.
e.g. Target number: 3 digits
Numbers provided: 2 digits
Multiple of: 12 
This will be impossible when the numbers provided are:
1 & 1
1 & 7
3 & 5
5 & 9
7 & 7
It may be tempting for students to suggest that whenever the interactivity provides two odd numbers, it will be impossible to create a three-digit multiple of 12. This is not the case; it is possible to create three-digit multiples of 12 with all other combinations of two odd numbers. Students could be asked to find which pairs of odd numbers can, and which cannot, create three-digit multiples of 12.
Finally, the "something else to think about" at the end of the problem could be used as a homework task or as a final challenge.

Key questions

How do you know if a number is a multiple of 3?
How do you know if a number is a multiple of 4?
How do you know if a number is a multiple of 6?
How do you know if a number is a multiple of 12?
How do you know you have found the biggest possible number?


Possible support

Some students may benefit from starting with two-digit target numbers and focusing on multiples of 2, 5 and 10. This more accessible context will still require students to reason and justify. As they become more confident, they can move on to multiples of 20, 4, 3...


Possible extension

Students could be encouraged to read the article about Divisibility Tests.
Once students are expert at single-digit multiples, they can be challenged to become expert at a selection of two-digit multiples. 

American Billions is an engaging extension activity which uses similar ideas to the ones met in this problem.