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# Counting

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How many sequences of seven positive integers \(a_1 are there such that each is less than \(100\) and each number apart from the last is a factor of the next number in the sequence?

Jul 9, 2022

#1
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If you are too lazy to do that, it suggests to me that you will be too lazy to try and understand anybodies answer.

Jul 9, 2022
edited by Melody  Jul 9, 2022
#2
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I'm sorry, I don't know what happened. When I was posting it the latex was working perfectly fine!

Guest Jul 9, 2022
#3
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How many sequences of seven positive integers \(a_1 are there such that each is less than 100 and each number apart from the last is a factor of the next number in the sequence?

Jul 9, 2022
#5
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How many sequences of seven positive integers a1 < a2 < a3 < a4 < a5 < a6 < a7 are there such that each is less than 100 and each number apart from the last is a factor of the next number in the sequence?

Jul 9, 2022
#6
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ok, I have not answered this yet but I will share my thoughts with you, maybe we can nut it out together.

The first thing I thought of it that I want the highest possible value of a_1.

Each number is different so the smallest factor possible is 2

so   if I let the the first one be A then the numbers will be

A,2A,4A,8A,16A,32A,64A

If A is 1 that is ok

If A is 2 then 64A is too big so the first term has to be1

The multiple could be 2 every time giving one possible sequence.

What about if one of the muliples was 3 rather than 2 then it would end in 96A,  So there are more possibilities there.

What about if one of the multiples 4 rather than 2 then it would end in 128A which is too big

so now you have a lot less possibilities to consider.

Can you find the answer and tell us what it is ? Jul 9, 2022