Multiple surprises

Why do this problem?

This problem offers a great opportunity for students to take the important mathematical step of starting from particular cases and moving towards generalisations. In exploring each of the 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

This printable worksheet may be useful: Multiple Surprises. This problem featured in an NRICH Secondary webinar in October 2020.

Ask students the first question:

"Can you find three consecutive numbers where the first is a multiple of 2, the second is a multiple of 3 and the third is a multiple of 4?"

Give them some time to think and then ask "Can you find another set? Can you find a set that you don't think anyone else in the class will have thought of?" If students have whiteboards they could write a set on their boards and hold them up. You could collect sets of three numbers that satisfy the constraints on the board for everyone to see.

"What do you notice about all the sets we've found?" Give students time to think on their own and to discuss with a partner, before discussing it with the whole class. Students might notice that the middle number is always an odd multiple of 3, and that there is a gap of 12 between one set and the next.

You could also invite students to explain how they found examples - this is a good opportunity to emphasise the importance of working systematically when tackling a maths problem.

Then set the next two challenges and invite students to work on them in a similar way:

"What if the first is a multiple of 3, the second is a multiple of 4 and the third is a multiple of 5?"

"What if the first is a multiple of 4, the second is a multiple of 5, and the third is a multiple of 6?"

Once they have had a chance to find some examples and make some conjectures, bring the class together to share findings, and discuss any conjectures: "When we were looking for multiples of 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 try the other challenges: 

"Is there a quick way to find sets of four consecutive numbers which are multiples of 2, 3, 4 and 5 (in this order)? Or five consecutive numbers which are multiples of 2, 3, 4, 5 and 6 (in this order)?"

Finally, the last challenge offers students the opportunity to show that they have really understood the structure of the problem:

"Can you use what you have discovered to help 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?"

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.