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Counting Factors

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

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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.


Age 7 to 14 Challenge Level:

Do you know a quick way to check if a number is a multiple of two?
How about three, four or six?
... View the problem


Why do this problem?

This problem offers a twist on the usual way of thinking about divisibility tests - rather than checking to see if a number is divisible by 2, 3, 4 or 6, students are asked to construct numbers that meet the necessary criteria.

Possible approach

Introduce the problem by showing the interactivity with multiples of two selected. 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. Challenge the class to develop a strategy for ALWAYS finding the highest even number on the first attempt.
Repeat the activity selecting multiples of 3 - this is a good opportunity to introduce the standard divisibility test for multiples of 3 if it has not been met before.
"Your next challenge is to develop a similar strategy for finding the highest multiple of 4, and the highest multiple of 6, on the first attempt."
In a computer room, pairs of students can work together at a 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.
Once the class have had time to work on the problem and develop and test their strategies, bring the class together. There are many ways in which this could be done; one possibility is to challenge each pair to construct the solution to a new problem generated by the interactivity.
"Will we be able to get all the way round the class without anyone making a mistake?"
Finally, the "something else to think about" at the end of the problem could be used as a homework task or in a follow-up lesson.
The article Divisibility Tests offers clear explanations of the various divisibility rules and why they work.

Key questions

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 you have found the biggest possible number?

Possible extension

Students could be encouraged to read the article about Divisibility Tests.
Ask students to imagine that the interactivity also asks them to construct multiples of 12. Challenge them to come up with a strategy and identify situations when a multiple of 12 cannot be constructed.

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

Possible support

The interactivity in the problem Largest Even offers students practice in finding even numbers using just two digits. In a similar way, the rest of the challenges in this problem could be adapted to two-digit numbers.