Problem
NT-B1-M09-P021 Smallest with \(36\) Divisors
#21
★★★★☆ Level 4 of 5
Find the smallest positive integer with exactly \(36\) positive divisors.
Compare the main factorisations of \(36\): \(9\cdot4\), \(6\cdot6\), \(6\cdot3\cdot2\), \(4\cdot3\cdot3\), \(3\cdot3\cdot2\cdot2\).
Candidates are \(2^8\cdot3^3=6912\), \(2^5\cdot3^5=7776\), \(2^5\cdot3^2\cdot5=1440\), \(2^3\cdot3^2\cdot5^2=1800\), \(2^2\cdot3^2\cdot5\cdot7=1260\). The smallest is \(1260\). Other patterns with a larger single exponent are even larger. Answer: \(1260\).
Level 4: several cases and optimisation.