Problem
NT-B1-M09-P015 Smallest with \(24\) Divisors
#15
★★★☆☆ Level 3 of 5
Find the smallest positive integer with exactly \(24\) positive divisors.
Check factorisations of \(24\) into factors \(a_i+1\).
Important candidates are \(2^7\cdot3^2=1152\), \(2^5\cdot3^3=864\), \(2^5\cdot3\cdot5=480\), \(2^3\cdot3^2\cdot5=360\), \(2^2\cdot3\cdot5\cdot7=420\). The smallest is \(360\), and the remaining patterns with larger exponents are larger. Answer: \(360\).
This is genuine olympiad optimisation, still accessible.