2026/068) What is the smallest positive integer that has exactly 24 factors

Basis

To find the number of positive divisors of an integer, find its prime factorization, add one to each exponent, and multiply those sums together.

That is If $N=p_1^{q_1}p_2^{q_2}\cdots p_n^{q_n}$

it shall have $(q_1+1) (q_2+1)\cdots(q_n+1)$ factors

kindy note that $p_1,p_2\cdots p_n$ are relatively prime and not $q_1+1,q_2+1\cdots q_n+1$ 

Solution

Let is factor 24 in as many ways as we can and put N as product of power of primes and for the number to be lower higher power shall be with lower numer

 24 = 24 this gives number $2^23$

24 = 8 * 3 giving $2^7 *3^2= 1152 $=  this is smaller

can we make it smaller 

24 = 4 * 2 *3 = 4 * 3 * 2 giving $ 2^3 *3^2 * 5 = 360$ 

if we try to make a smaller number we get 3 * 2 * 2 * 2 giving $2^2 * 3 * 5 * 7 = 420$ and it is larger

so the ans is $360$