# Thread: composite positive integers

1. ## composite positive integers

determine all composite positive integers n for which it is possible to arrange all divisors of n that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.

2. Originally Posted by nh149
determine all composite positive integers n for which it is possible to arrange all divisors of n that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.

$\displaystyle n = p^k, k>2$

3. A proof would be nice; I think that's what the problem is asking for.

Edit : 9 3 6 12 2 4 18 36 is a counter-example to your "solution".

4. Originally Posted by Bruno J.
A proof would be nice; I think that's what the problem is asking for.

Edit : 9 3 6 12 2 4 18 36 is a counter-example to your "solution".
Could you tell me what composite number you are using; that is, what prime you are raising to what power?
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5. Those are the divisors of 36. If you join them in a circle (so that 9 and 36 are adjacent), you will see that the criterium of the problem is satisfied, and 36 is not a prime-power.

(2 4 6 3 12) is another counter-example, with the divisors of 12.

6. Originally Posted by Bruno J.
Those are the divisors of 36. If you join them in a circle (so that 9 and 36 are adjacent), you will see that the criterium of the problem is satisfied, and 36 is not a prime-power.

(2 4 6 3 12) is another counter-example, with the divisors of 12.
Sorry to be difficult, but I still cannot grasp how you can generate 12 or 36 from raising a prime number to a power.

I'm at a loss here.

Could you please show me how you can take a prime number and raise it to a power and come up with 12 or 36? Please.

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I think I see what you are implying:
BOTH p & k MUST be prime numbers:

$\displaystyle n = p^k$

7. Are you feeling alright?

8. Originally Posted by Bruno J.
Are you feeling alright?
I feel just fine.

Could you show how it is possible to raise a prime number to any power and come up with 12 or 36?

.

9. Originally Posted by aidan
I feel just fine.

Could you show how it is possible to raise a prime number to any power and come up with 12 or 36?

.

It is not possible. Why do you keep on asking? Is it that you meant in your answer p^k , k > 2, that p is a prime? And if you did why didn't you write so?
And anyway: can you prove your claim?

Tonio

10. Originally Posted by aidan
Sorry to be difficult, but I still cannot grasp how you can generate 12 or 36 from raising a prime number to a power.

I'm at a loss here.

Could you please show me how you can take a prime number and raise it to a power and come up with 12 or 36? Please.
? You are the only one who has said anything about raising a prime to a power. All that is being asked about here are composite positive integers. Bruno J's counter example was specifically to show that you do NOT have to have $\displaystyle p^k$.

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I think I see what you are implying:
BOTH p & k MUST be prime numbers:

$\displaystyle n = p^k$