Smallest Positive Integer N for which tow(n)=6 is True

[kathrynag]

Find the form of all n in N satifying tow(n)=6. (sorry don't know how to write this Tex). What is the smallest positive integer n for which this is true?

tow(n)=6 so 6=(1+a1)(1+a2)----(1+ak) where n=p1^a1p2^a2---pk^ak

 

[kathrynag]

\tau(n)=6
We have \tau(n)=(1+\alpha_{1})(1+\alpha_{2})...(1+\alpha_{k}) for k=p^{\alpha_{1}}p^{\alpha_{2}}...p^{\alpha_{k}}
So 6=(1+\alpha_{1})(1+\alpha_{2})...(1+\alpha_{k})
I just don't knw about the next step...

 

[Dick]

There is no standard next step. You have to think about it. Can you find any examples of integers n such that tau(n)=6? You should be able to find a lot. Use those examples to make a guess. Now try and find an argument to prove your guess is correct. This is really a basic creative math question. There's no cookbook method.

 

[kathrynag]

Well 2^5 works
(5+1)=6
3^1*7^2
(1+1)(2+1)=2*3=6

So n=p^5 or n=p^2p^1

 

[kathrynag]

n=2^2*3=4*3=12 smallest possible n?

 

[Dick]

Well 2^5 works
(5+1)=6
3^1*7^2
(1+1)(2+1)=2*3=6

So n=p^5 or n=p^2p^1

Now that's a good start. Yes, 32=2^5 works. And 3^1*7^2=147 works. Any tentative conclusions so far?

 

[kathrynag]

I think it's something of the form n=p^5 or n=p^2*p
I think those are the only forms that work

 

[Dick]

I think it's something of the form n=p^5 or n=p^2*p
I think those are the only forms that work

I would write that as p^5 or p1^2*p2. There are two different primes there. And sure, you can only factor 6 as 6*1 or 2*3. Now keep going.

 

[Dick]

n=2^2*3=4*3=12 smallest possible n?

Sure it is. Now can you prove it? I checked your page once and you want to be a mathematician, right? Mathematicians prove things. This an easy one. Try an omit the '?', by giving me an argument why it's true.