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
On
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.