Dozens

Age 7 to 14

Challenge Level Yellow star

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.

You may be interested in watching the NRICH team discussing the use of this problem in the recording of the March 2021 NRICH secondary teacher webinar (this task begins at about 4:30), and/or the recording of the team presenting the problem to students at the Cambridge Festival. This problem also featured in an NRICH Primary webinar in April 2021.

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.

1 & 4 are a good choice of digits. The largest possible three digit numbers change as the multiples change:

914 largest $2x$

741 largest $3x$

714 largest $6x$

416 largest $4x$

144 largest $12x$

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

Here are the three-digit numbers that can be made with the other pairs of odd numbers:

1 & 3 --> 312
1 & 9 --> 912
3 & 7 --> 732
5 & 5 --> 552
7 & 9 --> 972

1 & 5 --> 516
3 & 3 --> 336
3 & 9 --> 936
5 & 7 --> 756
9 & 9 --> 996

You may want to draw attention to the properties of the three different sets of pairs of odd numbers:

The five pairs of odd numbers which are impossible for multiples of 12 all add up to something which is equal to 2 mod 3, which means that the third digit has to be 1, 4, or 7. It must therefore be 4 so that the number is even, but all the two digit multiples of 4 ending in 4 have an even tens digit as well (04, 24, 44, 64, 84).

There are ten more pairs of odd numbers, five of which sum to 1 mod 3, and so the third number can be 2, 5, or 8, and all of 12, 32, 52, 72, 92 are multiples of 4.

The last five pairs add up to a multiple of 3, so the last digit can be one of 0, 3, 6, 9. We have 16, 36, 56, 76, 96, all multiples of 4!

This argument can be used to show that it is always possible to find a four-digit multiple of 12 given two odds and an even:
If the three digits add to 2 mod 3, add a 4 and put the evens digit in the 10s column.
If the three digits add to 1 mod 3, add a 2 and put one of the odds in the 10s column, or add an 8 and the evens digit in the 10s column.
If the three digits add to 0 mod 3, add a 0 and put the even number in the 10s column, or add a 6 and put an odd digit in the 10s column.

You won't be able to find a four-digit multiple of 12 if the three digits are all odd and sum to 2 mod 3 (e.g. 3, 3, 5).

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.

Students could also be asked to consider four-digit multiples of 12 when they are given two digits and/or three digits. For example, can they create a four-digit multiple of 12 when they are given 3, 3 and 5?

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.

For slightly older students, Common Divisor offers an intriguing follow-up problem.

Number and algebra

Geometry and measure

Probability and statistics

Working mathematically

Advanced mathematics

For younger learners

Dozens

Why do this problem?

Possible approach

Key questions

Possible support

Possible extension

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