A number has 24 factors.The multiplication of its prime factors is 30(taking 1 at a time).What is the ratio of the square roots of the highest and the smallest such number possible?

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- Jun 23rd 2009, 08:24 AMjashansinghala difficult problem
A number has 24 factors.The multiplication of its prime factors is 30(taking 1 at a time).What is the ratio of the square roots of the highest and the smallest such number possible?

- Jun 23rd 2009, 09:23 AMSwlabr

The hard part is the first sentence. You know the primes that divide such a number (as these are the primes that multiply to get 30), and the powers of the primes sum to 24 and are all non-zero (as otherwise the product would not be 24). So, what is the largest number possible? What is the smallest?

Big hint: let be such a number, with . Then the largest is the one with and .

(Unless I'm missing something here?) - Jun 23rd 2009, 09:56 AMTheAbstractionist
I took

**jashansinghal**’s first line to mean 24 factors, not 24 prime factors. But then I’m unable to find any integer with exactly 24 factors. Maybe I’m missing something as well. (Worried) - Jun 23rd 2009, 10:01 AMSwlabr
- Jun 23rd 2009, 12:26 PMMoo
Hello,

For example, 6 has 4 factors : 1,2,3,6

If I'm not mistaking (hasn't seen this for a while) : the number of factors of a number is

So here, we have

And the product of its prime factors is 30 :

Then, we have to be able to find the smallest and greatest possible numbers responding to these conditions... So I'm pretty sure we have to study the prime decomposition of 30 :D

But now... I have to take a shower (Rofl) - Jun 23rd 2009, 12:59 PMPlato
Each of these has 24 factors: .

If I understand the bit about the product being 30 then they are smallest and largest. - Jun 23rd 2009, 01:05 PMTheAbstractionist
- Jun 23rd 2009, 01:43 PMMoo
and satisfies all the conditions...

- Jun 23rd 2009, 02:09 PMPlato
- Jun 23rd 2009, 02:27 PMMoo
- Jun 24th 2009, 02:25 AMjashansinghal
- Jun 24th 2009, 11:44 AMTheAbstractionist
All right guys, I just remembered this: the tau function which gives the number of factors of each integer has the following property: if where the are distinct primes, then

So for this particular problem, the distinct primes are 2, 3, and 5 and the number is of the form where The possibilities are thus

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

The largest number is therefore and the smallest is Hence the answer to the question is - Jun 24th 2009, 12:41 PMMoo
- Jun 24th 2009, 03:25 PMTheAbstractionist