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14 Divisors

What is the smallest number with exactly 14 divisors?

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Repeaters

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.

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Oh! Hidden Inside?

Find the number which has 8 divisors, such that the product of the divisors is 331776.

Legs Eleven

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem?

This problem involves a significant 'final challenge' which can either be tackled on its own or after working on a set of related 'building blocks' designed to lead students to helpful insights. At one level it provides self-checking practice of the algorithms for column addition and short division. At its heart it is about challenging and developing students' understanding of place value.

Initially working on the building blocks then gives students the opportunity to work on harder mathematical challenges than they might otherwise attempt.

The problem is structured in a way that makes it ideal for students to work on in small groups.

Possible approach

Hand out a set of building block cards (Word, PDF) to each group of three or four students. (The final challenge will need to be removed to be handed out later.) Within groups, there are several ways of structuring the task, depending on how experienced the students are at working together.

Each student, or pair of students, could be given their own building block to work on. After they have had an opportunity to make progress on their question, encourage them to share their findings with each other and work together on each other's tasks.

Alternatively, the whole group could work together on all the building blocks, ensuring that the group doesn't move on until everyone understands.

When everyone in the group is satisfied that they have explored in detail the challenges in the building blocks, hand out the final challenge.

The teacher's role is to challenge groups to explain and justify their mathematical thinking, so that all members of the group are in a position to contribute to the solution of the challenge.

It is important to set aside some time at the end for students to share and compare their findings and explanations, whether through discussion or by providing a written record of what they did.

Key questions

What important mathematical insights does my building block give me?
How can these insights help the group tackle the final challenge?

Possible extension

Of course, students could be offered the Final Challenge without seeing any of the building blocks.
After establishing a general proof, students might like to try some similar problems:
Double Digit
Special Number
Repeaters

Possible support

Encourage groups not to move on until everyone in the group understands. The building blocks could be distributed within groups in a way that plays to the strengths of particular students.

For students who have a weak grasp of place value, Diagonal Sums gives addition practice in a context where the place value structure is quite explicit.