# Smallest value of n given its sixth divisor

• Mr Davis 97
In summary, the smallest possible value of n is 60, where the divisors are in the order of 1, 2, 3, 4, 5, 6, 10, 12, 15. However, it is also possible for n to equal 1800, where the divisors are in the order of 1, 3, 5, 7, 9, 11, 15. To find the minimal solution, it is necessary to use pairwise relatively prime numbers, such as in the example of 1, 3, 5, 7, 11, 15.
Mr Davis 97

## Homework Statement

All of the divisors of ##n## are in increasing order: ##1=d_1 < d_2 < \dots < d_t = n##. We know that ##d_6=15##. What is the smallest possible value of ##n##?

## The Attempt at a Solution

Here is my reasoning. We have the chain ##1 < d_2 < d_3 < d_4 < d_5 < 15 < n##, where we make ##15## the largest factor that's not ##n##. Since ##15~|~n##, we have that ##5~|~n## and ##3~|~n##. Hence, we have to put ##3## and ##5## somewhere. The minimal sequence is then ##1 < 2 < 3 < 4 < 5 < 15 < n##, so ##n=2\cdot 3\cdot 4 \cdot 5 \cdot 15 =1800##

Why not ##60##? ##1,2,3,4,5,15 \,|\,60## and ##t=6##.

Edit: Doesn't work, since ##6,10## and ##12## are not listed. But this applies to your example, too.

fresh_42 said:
Why not ##60##? ##1,2,3,4,5,15 \,|\,60## and ##t=6##.

Edit: Doesn't work, since ##6,10## and ##12## are not listed. But this applies to your example, too.
I'm not seeing what you mean. Where does my logic go bad?

Mr Davis 97 said:
I'm not seeing what you mean. Where does my logic go bad?
Mr Davis 97 said:
All of the divisors of ##n## are in increasing order: ##1=d_1 < d_2 < \dots < d_t = n##. We know that ##d_6=15##.
So in case of ##n=60## as well as ##n=1800## we have ##1=d_1<2=d_2<3=d_3<4=d_4<5=d_5<6=d_6<10=d_7<12=d_8<15=d_9\neq d_6##.
In case of ##n=1800## the list also includes ##d_7=8## and ##d_8=9## which shifts the indices even further.

fresh_42 said:
So in case of ##n=60## as well as ##n=1800## we have ##1=d_1<2=d_2<3=d_3<4=d_4<5=d_5<6=d_6<10=d_7<12=d_8<15=d_9\neq d_6##.
In case of ##n=1800## the list also includes ##d_7=8## and ##d_8=9## which shifts the indices even further.
I see. Any idea on how to proceed then?

I tried ##2,3,4##. But then we have ##2\cdot 3=6## and ##2\cdot 4=8## and ##3\cdot 4=12##, too. These are already six numbers smaller than ##15##. Besides that, with ##15## we have an automatic divisor ##5##. So ##1,3,5,15## are set. Etc...

What about if we have 1,3,4,5,12,15?

You cannot have ##4## without ##2##.

fresh_42 said:
You cannot have ##4## without ##2##.

Mr Davis 97 said:
Then you need a 9: 15*3=45. I think you need to use pairwise relatively-prme numbers.

Mr Davis 97 said:
Looks good, but I'm not sure whether this is already the minimal solution.

fresh_42 said:
Looks good, but I'm not sure whether this is already the minimal solution.
Look at my post #10. I think he needs to use pairwise relatively prime.

WWGD said:
Look at my post #10. I think he needs to use pairwise relatively prime.
That wasn't a valid objection, because we do not have to use a prime twice. ##3\,|\,n## and ##15\,|\,n## doesn't make ##9## a divisor. I even think that we can substitute ##11## by ##9## in his example.

Mr Davis 97 said:
I see. Any idea on how to proceed then?
You know 3 and 5 must appear in the list.
Next question should be whether 2 can appear. If it does, what else must come before 15?

## What is the "Smallest value of n given its sixth divisor" problem?

The "Smallest value of n given its sixth divisor" problem is a mathematical problem that asks for the smallest positive integer n that has a particular number as its sixth divisor. This problem is often used as an example in number theory and can also be used as a coding challenge.

## How is the "Smallest value of n given its sixth divisor" problem solved?

The "Smallest value of n given its sixth divisor" problem can be solved by finding the prime factorization of the given number and then using the formula n = p^5, where p is the largest prime factor of the given number. This formula ensures that n has the given number as its sixth divisor.

## What is the significance of the "Smallest value of n given its sixth divisor" problem?

The "Smallest value of n given its sixth divisor" problem has significance in number theory and can be used to demonstrate the relationship between prime numbers and their divisors. It also has practical applications in coding and can be used to optimize algorithms for finding divisors of a given number.

## Are there any variations of the "Smallest value of n given its sixth divisor" problem?

Yes, there are variations of the "Smallest value of n given its sixth divisor" problem, such as finding the smallest value of n given a different divisor or finding the smallest value of n that has a certain number of divisors. These variations can be solved using different formulas and techniques.

## Can the "Smallest value of n given its sixth divisor" problem be solved for non-integer numbers?

No, the "Smallest value of n given its sixth divisor" problem is typically only solved for positive integers. However, there are variations of the problem that involve finding the smallest value of n for non-integer divisors, such as finding the smallest value of n that has a certain irrational number as its divisor.

• Calculus and Beyond Homework Help
Replies
3
Views
691
• Calculus and Beyond Homework Help
Replies
14
Views
881
• Calculus and Beyond Homework Help
Replies
10
Views
674
• Precalculus Mathematics Homework Help
Replies
1
Views
880
• General Math
Replies
4
Views
1K
• Precalculus Mathematics Homework Help
Replies
3
Views
2K
• Calculus and Beyond Homework Help
Replies
4
Views
857
• Calculus and Beyond Homework Help
Replies
2
Views
3K
• Precalculus Mathematics Homework Help
Replies
4
Views
875
• Introductory Physics Homework Help
Replies
12
Views
899