ISL 2004-N1

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sourav das
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ISL 2004-N1

Unread post by sourav das » Sun Apr 01, 2012 1:47 am

Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers a such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.
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*Mahi*
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Re: ISL 2004-N1

Unread post by *Mahi* » Sun Apr 01, 2012 3:24 pm

Intuition:
We generally try putting a prime/prime power at first wherever we see "there exist infinitely many positive integers such that..." ;)
Next Hint:
Let $a=p^k$. Then to find $k$, let $\gcd (a,p)=1$. So $n=\tau(p^kn)=\tau(p^k)\tau(n)=(k+1)\tau(n)$.
Can you guess now what might $k$ be?
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