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Find the smallest positive integer with exactly 30 positive factor

First, I use function $\tau$ to find the exponential that gives $2×3×5$ and I want to find the smallest value. How can I find it use inequality to help

And how to find the value of $\sqrt{8×13×15×17+49}$? I get $x=8$ and doing perfect square ... but stuck who can help me, please

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    $\begingroup$ Your question is a bit unclear. Can you give some examples of the smallest value of a positive number with 2,3,4 positive factors? What do you mean with positive factors? $\endgroup$ – kvantour Jan 31 at 13:11
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    $\begingroup$ What have you tried? Given the primne factorization $n=\prod p_i^{a_i}$ , do you know how to compute the number of factors of $n$? $\endgroup$ – lulu Jan 31 at 13:11
  • $\begingroup$ @kvantour To me these feel like two separate questions that should have been asked separately, in my opinion. $\endgroup$ – Robert Soupe Jan 31 at 18:39
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The number of factors of a natural is given by the product of the multiplicities of its prime factors plus one, $(m_2+1)(m_3+1)(m_5+1)\cdots$.

To obtain $30=2\cdot3\cdot5$ factors, you need multiplicities $1,2$ and $4$, which you will assign to the smallest possible primes, by decreasing order

$$5^13^22^4=720.$$

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What do you mean by "exactly 5 positive factors"? Are these distinct or can they be the same? If the same then the smallest such integer is $2^5= 32$. It they must all be different then it is 2(3)(4)(5)(6)= 720.

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  • $\begingroup$ The question about distinctness doesn't make sense. $1$ has thirty factors $1$, and many more. $\endgroup$ – Yves Daoust Jan 31 at 13:22
  • $\begingroup$ Because of that "1" is not usually considered a factor. $\endgroup$ – user247327 Jan 31 at 13:33
  • $\begingroup$ Don't you understand ? $2$ has thirty factors $2$, and many more. $\endgroup$ – Yves Daoust Jan 31 at 15:34

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