Factors and Multiples Chain

Choose a starting number from a 1-100 square and cross it out.

Then choose a factor or multiple of that number.

Keep crossing out factors or multiples of the last number in the chain.

For example, Charlie started with 60, 30, 6, 96, 16, 32, 8, 56, 7, 21, 42,...

What's the longest chain you can make?

You may wish to download a 1-100 square to work on, or you could use the interactivity below.

 

 

 

 

We'd love you to share how you've tried this and what you've discovered. You can add a comment below, or you can email us - your work may be featured in the showcase.

Comments

I really enjoy ... still learning and understanding how it goes.

Through trial and error with 7th grade students (13 years old) we have found that 74 is the maximum chain that can be made.

We are still thinking, but we believe that the reason has to do with the 26 primes in between 1 and 100 (100-26 = 74).

Have you been able to produce a chain of 74? Do share it if you have.

By the way, if your theory is right, you may be able to produce a chain of 75 - there are just 25 primes in between 1 and 100.

Yes - if you've linked all you can, except for the 1, you can link that to a prime number.

I have!!! Luckily

I am almost certain the maximum chain is between 77 and 78 because I have made a chain that is 77 numbers long.

My chain is:

68, 34, 17, 85, 5, 95, 19, 38, 76, 4, 92, 46, 23, 69, 3, 87, 29, 58, 2, 62, 31, 93, 1, 39, 78, 26, 52, 13, 91, 7, 77, 11, 99, 33, 66, 22, 44, 88, 8, 56, 28, 84, 42, 21, 63, 9, 81, 27, 54, 18, 90, 45, 15, 75, 25, 50, 100, 20, 40, 80, 16, 64, 32, 96, 6, 36, 72, 24, 48, 12, 60, 30, 10, 70, 14, 98, 49

 

I think it may be possible to produce a chain of 78 because I found a pattern that is consistent and it predicted a length of 78 however I spent a lot of time rearranging the chain and I still haven't found a way to make it 78 long.

That's a very impressive chain, well done!

You mentioned a consistent pattern that you found, could you explain what that pattern is?

high score 23

Can you list the numbers in your chain?

My longest chain is 41 numbers! I don't know any more combinations there are for it because I've forgotten how I did my chain! OOOPS!!! Silly me! I'm thinking of doing it again with number 1-121 on an 11 x 11 grid! Do you think I have a good idea for expanding minds?

Do let us know how you get on with the 1-121 grid, and do post the list of numbers in your chain.

My chain is:

77, 11, 121, 1, 22, 44, 88, 8, 104, 57, 3, 30, 90, 9, 45, 5, 50, 100, 10, 20, 40, 80, 4, 48, 96, 6, 12, 36, 108, 54, 18, 72, 24, 2, 42, 21, 7, 14, 28, 56 and 112.

 

41 numbers altogether!

Hey, I've got two questions that will put your brain in gear!

1. What do all the 41 numbers make when all added up?
2. What do all the DIGITS added up make? Share your answers!

That's a good first chain, but I'm afraid you've made a mistake along the way - you're not allowed to go from 104 to 57.

How about having another go and trying to make an even longer chain?

There are lots of numbers you haven't used yet which could be added to your current chain:
60 could go between 10 and 20
15 could go between 30 and 90
120 between 20 and 40...

You could also remove 77 from the start and replace it with 99, 33 and 66.

You could also change the order of some of your numbers.
For example, instead of 2, 42, 21, 7, 14, 28, 56 and 112, as you have at the end you could have:
2, 112, 56, 28, 14, 42, 21, 7... and no longer be stuck!

Of course, you could make a fresh start...

We got all the numbers. Has anyone else?

I'm afraid it's not possible to make a chain that uses all the numbers.

How did you manage to include all the large prime numbers (e.g. 53, 59, 61, 67, 71, 73 ...) in your chain?

We are Year 5 and our longest chain is 51. We started on 99.

Well done - it's not easy to produce a chain of 51 numbers.

Could you tell us which numbers you included in your chain?

Here is my chain:

96, 48, 24, 12 , 6, 18, 36, 72, 3, 84, 42, 21, 63, 7, 77, 11, 1, 13, 65, 5, 10, 60, 20, 40, 80, 2, 56, 28, 4, 8, 16, 32, 64


96 and 64 had the most prime factors so I worked on both sides of the chain until I was stuck and connected them both with a 1. I made sure that there were no duplicates.

Thank you for sharing your chain.

You could make it even longer...

e.g. 1: you could add 88, 44, 22, 66 and 33 after the 11 (and before the 1)

e.g. 2: you could add 25, 50 and 100 between the 5 and the 10

Perhaps you can find some other numbers to add to your chain...

I think this may be right

52, 26, 78, 39, 13, 91, 7, 49, 98, 14, 70, 10, 80, 40, 20, 100, 50, 25, 75, 15, 60, 30, 90, 45, 9, 81, 27, 54, 18, 72, 36, 12, 24, 6, 48, 16, 64, 32, 96, 4, 92, 46, 23, 69, 3, 93, 31, 62, 1, 63, 21, 42, 84, 28, 56, 8, 88, 44, 22, 66, 33, 99, 11, 55, 5, 85, 17, 34, 68, 2, 38


I need just 3 more do you guys have any idea of what numbers could be added to make 74?
I did this at school and got the best in the class any help ??????????

The problem I see with getting a complete chain is the primes, of course. More importantly, the primes over 50.

If you start your chain with 53, the next multiple is 106, too high, so you have to find a factor. The only factor is 1. The same is true for all the primes above 50. At most you can use one of them in your chain, at one end.

For any N, you can only have one prime larger than (N/2) in your chain.

examples
N=1: 1
N=2: 1, 2
N=3: 3,1, 2
N=4: 4, 2, 1, 3
N=5: oops, can't use both 3 and 5. Max is chain of 4 numbers 5, 1, 2, 4 or 3, 1, 2, 4
N=6: 4, 2, 6, 3, 1, 5
N=7: can't use both 5 and 7. Max is chain of 5 numbers.

There are 10 primes above 50 ( 53, 59, 61, 67, 71, 73, 79, 83, 89, 97) so there is an absolute max chain length of 91 (100-10+1). I doubt that can actually be achieved though.

There is probably a similar problem with the primes between N/2 and N/3, being also limited in usage (like you can only use 2 or 3 of them, max), but I haven't given it much thought yet.

We did:

100, 50, 25, 5, 10, 20, 40, 80, 8, 16, 4, 24, 48, 96, 32, 64, 2, 98, 49, 1, 22, 44, 88, 11, 99, 33, 66, 6, 12, 36, 72, 3, 9, 18, 54, 27, 81.

That's a great start, but even though you can't add numbers at the start or at the end of your chain, you can make your chain longer by adding numbers in between some of your numbers.

For example:
you could add 75 between the 25 and 5
you could add 15, 30 and 60 after the 5 and before the 10
you could add 7, 42, 21 and 63 between the 49 and 1

Can you find any more numbers that can be added to your chain?

I got 63! It was really challenging getting from about 57 to 63. Had to get out the pencil and paper. I don't know if I can go higher right now.

I love this activity, but wish solutions were not posted where students can so obviously see them as it discourages them from thinking themselves, but just copying. Could the interactive be posted elsewhere for students to use with no solutions shown?

Thanks

ajk44's picture

Hi Mike, you can access the interactive on its own without comments here: https://nrich.maths.org/factmult/ Hope that helps!

Thanks so much! Great activity

I have successfully made it to 46 and have done this starting at 100 and then I have divided then I have multiplied numbers to try to get the order.
The order of the numbers are

100, 50, 25, 5, 10, 20,40, 80, 16, 32, 64, 4, 12, 24, 48, 96, 6, 18, 36, 72, 9, 27, 81, 3, 15, 45, 90, 30, 60, 2, 14, 56, 8, 88, 11, 44, 22, 66, 33, 99, 1, 7, 21, 42, 84, 28.

high score 23

That's a good first chain! Can you tell us what numbers you used in your chain?

Here´s my chain:

52, 26, 78, 13, 91, 7, 49, 98, 14, 70, 10, 80, 40, 20, 100, 50, 25, 75, 15, 60, 30, 90, 45, 9, 81, 27, 54, 18, 72, 36, 72, 36, 12, 24, 6, 48, 16, 64, 32, 96, 4, 92, 46, 23, 69, 31, 62, 1, 63, 21, 42, 84, 28, 56, 8, 88, 44, 22, 66, 33, 99, 11, 55, 5, 85, 17, 34, 68, 76, 19, 57.


That´s it! please feel free to show to your parents or kids, siblings, and family! Also feel free to copy too! Have fun trying to beat me!

Also can you guys tell me how to get a scale higher than 100?!

Thanks and have fun!

That's an excellent chain, but I'm afraid you've made a few errors. I've hidden them below in case you want to find them yourself:

- You've repeated 76 and 32, in "76, 32, 76, 32" so remove the extra copy of those.
- At "69, 31, 62, 1", 31 isn't a factor of 69. However, if you remove 31 and 62 from there, then that fixes the gap in your chain.
- At the end, 76 is not a factor or multiple of 68, nor are 19 or 57. However, if you put 2 between 68 and 76, then you can connect them.
- You could also put 3 at the end to add another number and then perhaps another multiple of 3.

With these changes, you could make a chain of 70.

As for a scale larger than 100, I'm afraid the interactivity only supports a 1-100 square. However, you could make a different grid of your own using a spreadsheet.

After one day trial the longest chain I got was 66:

 

If anyone has a longer chain please do share it!

Thank you

That is an excellent chain. Would it be possible to make it longer by inserting some numbers in between existing parts of your chain? For example, you could make it a little longer by inserting 98 between 7 and 14

Year 6 pupil proud of achieving a chain of 73. A proud teacher too :)

That's a very long chain. Could you list the numbers in your chain? Or alternatively, could you send us a screenshot of your number chain in the interactivity?

Here is my chain:

68, 34, 17, 85, 5, 95, 19, 38, 76, 4, 92, 46, 23, 69, 3, 87, 29, 58, 2, 62, 31, 93, 1, 39, 78, 26, 52, 13, 91, 7, 77, 11, 99, 33, 66, 22, 44, 88, 8, 56, 28, 84, 42, 21, 63, 9, 81, 27, 54, 18, 90, 45, 15, 75, 25, 50, 100, 20, 40, 80, 16, 64, 32, 96, 6, 36, 72, 24, 48, 12, 60, 30, 10, 70, 14, 98, 49

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