# Find the smallest value of positive integer !!

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

• 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? – kvantour Jan 31 at 13:11
• 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 $n$? – lulu Jan 31 at 13:11
• @kvantour To me these feel like two separate questions that should have been asked separately, in my opinion. – Robert Soupe Jan 31 at 18:39

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
$$5^13^22^4=720.$$
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.
• The question about distinctness doesn't make sense. $1$ has thirty factors $1$, and many more. – Yves Daoust Jan 31 at 13:22
• Don't you understand ? $2$ has thirty factors $2$, and many more. – Yves Daoust Jan 31 at 15:34