Multiple surprises
Sequences of multiples keep cropping up...
Problem
Multiple Surprises printable worksheet
Can you find a few sets of ten consecutive numbers in which:
- the first is a multiple of 1
- the second is a multiple of 2
- the third is a multiple of 3
- the fourth is a multiple of 4
- the fifth is a multiple of 5
- the sixth is a multiple of 6
- the seventh is a multiple of 7
- the eighth is a multiple of 8
- the ninth is a multiple of 9
- the tenth is a multiple of 10?
This is a hard problem, so you may want to watch the video below, in which Charlie and Liz discuss how mathematicians might approach a problem like this.
Alternatively, you can scroll down to get some guidance on how to get started on the problem.
Apologies - 9:30 minutes after the start, we mention adding 600, whereas of course we should have said "adding 540"!
When faced with a difficult problem, mathematicians sometimes simplify the problem, and as they work on the simplified version(s), try to gain some insights which might prove useful when they return to the original problem.
Here are some simpler problems involving consecutive numbers and multiples, which might help you prepare for tackling the original problem:
Can you find sets of three consecutive numbers where the first is a multiple of 1, the second is a multiple of 2, and the third is a multiple of 3?
Can you find several examples?
What do you notice? Have you any conjectures?
Do your conjectures apply in the following cases:
When the first number is a multiple of 2, the second is a multiple of 3 and the third is a multiple of 4?
When the first number is a multiple of 3, the second is a multiple of 4 and the third is a multiple of 5?
When the first number is a multiple of 4, the second is a multiple of 5, and the third is a multiple of 6?
How about sets of two consecutive numbers, where the first is a multiple of 9, and the second is a multiple of 10? Or sets of four consecutive numbers? Or...?
Can you use what you have discovered to help you generate several sets of numbers that satisfy the criteria in each of the cases you've tried?
Can you use what you have discovered to help you answer the original question?
With thanks to Don Steward, whose ideas formed the basis of this problem.
Teachers' Resources
Why do this problem?
This problem offers a great opportunity for students to take the important mathematical step of using particular cases and to move towards generalisations. In exploring each of the simpler challenges, students will get plenty of practice at working with multiples and lowest common multiples, which will deepen their understanding of factors and multiples and help them appreciate the significance of prime factorisation.
Possible approach
In the video that accompanies the problem (and in the recording above), Charlie and Liz are trying to model how you might interact with students in your class. You could watch the video(s) and adapt what you do according to the age range you are working with.
Alternatively, you could show the problem video to your class, pausing when recommended to offer some thinking and playing time, and to facilitate discussion.
Once students have had a chance to explore the simpler challenges, bring the class together to share findings, and discuss any conjectures. The conversation could go along the lines of: "When we were looking for multiples of 1, 2, 3 and 4, we added 12 to find the next set of solutions. When we looked at 3, 4 and 5, there was a gap of 60 between solutions, and there was also a gap of 60 when we looked at 4, 5 and 6. Can anyone explain why?"
Students may explain that they need to add a multiple of 3 to get another multiple of 3, a multiple of 4 to get another multiple of 4, and so on, so for every number in the set to work, they need to add a common multiple, and to find the next possible set they need the lowest common multiple. Once this insight has been understood, students can apply it to other examples of their choice and then return to the original problem of ten consecutive numbers.
Key questions
If I add the same number to a set of three consecutive numbers, will the new set of numbers be consecutive?
If I know that a number is a multiple of 3, what do I need to add to it to get another multiple of 3?
Which numbers are multiples of 2, 3 and 4?
Possible support
Encourage students to start with just a pair of constraints. For example, "Can you find two consecutive numbers where the first is a multiple of 3 and the second is a multiple of 4?" Then once they have found a family of examples, add the third constraint.
Possible extension
Invite students to consider how they can extend the last part of the question to generate a set of $n$ consecutive numbers so that the first is a multiple of 1, the second is a multiple of 2, the third is a multiple of 3 and so on. They might also like to consider how this relates to the idea that it's possible to find long sequences of consecutive numbers that do not contain any primes.
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