Find the least positive integer with $24$ positive divisors
If we assume the integer is $=a^x.b^y.....$
And $(x+1)(y+1)......=24$ Then there are too many variables How can I get least variables?
2026-03-25 06:01:29.1774418489
Find the least positive integer with 24 positive divisors
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For $n=p_1^{\alpha_1}\cdots p_k^{\alpha_k}$ the number of divisors is $(\alpha_1+1)\cdots(\alpha_k+1)=24$. The best case is to consider the $\alpha_k$s as decreasing when $p_i$s are increasing prime factors so some possible such cases are$$\alpha_1=23\\(\alpha_1,\alpha_2)=(11,1)\\(\alpha_1,\alpha_2)=(7,2)\\(\alpha_1,\alpha_2)=(5,3)\\(\alpha_1,\alpha_2,\alpha_3)=(5,1,1)\\(\alpha_1,\alpha_2,\alpha_3)=(3,2,1)\\(\alpha_1,\alpha_2,\alpha_3,\alpha_4)=(2,1,1,1)$$by a simple investigation the answer is $360$