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November 4th, 2017, 06:14 AM   #1
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Some problems in elementary arithmetic functions

Hello, I ran in to this problem and I cant seem to show this although it seems intuitively correct:

Given any positive integer $n>1$, prove that there are infinitely many integers $x$ satisfying $d(x)=n$. Where $d(n)$ is the number of positive divisors of $n$.

I will appreciate the help.
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November 4th, 2017, 07:57 AM   #2
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Consider the set of integers that consists of all primes raised to the first power. How many distinct divisors does each number in that set contain? How many numbers does that set contain?

Now consider the set of integers that consists of all primes raised to the second power. How many distinct divisors does each number in that set contain? How many numbers does that set contain?

Can you build a proof by induction from those clues?
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November 11th, 2017, 09:00 PM   #3
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That was very helpful! thank you, and yes I managed to solve it .
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