NT-B1-M09-P024 Smallest with \(60\) Divisors

Put exponents in decreasing order and compare multiplicative decompositions of \(60\).

To minimise the number, it is enough to consider \(2^{a_1}3^{a_2}5^{a_3}\cdots\), where \(a_1\ge a_2\ge a_3\ge\cdots\). We need \((a_1+1)(a_2+1)\cdots=60\). The main candidates are \(2^9\cdot3^2\cdot5=23040\), \(2^5\cdot3^4\cdot5=12960\), \(2^4\cdot3^3\cdot5^2=10800\), \(2^4\cdot3^2\cdot5\cdot7=5040\). Patterns with one or two very large exponents are even larger. The smallest candidate is \(5040\), and \(\tau(5040)=\tau(2^4\cdot3^2\cdot5\cdot7)=5\cdot3\cdot2\cdot2=60\). Answer: \(5040\).

Strong optimisation problem, suitable as the module finale.