Mathematics - Permutation & Combination Question with Solution | TestHub

$1440=2^5.3^2.5^1$

No. of divisors $=(5+1)\cdot (2+1)\cdot (1+1)=36$

Product of divisors $=\mathrm{1.2.3}\dots \dots .480.720.1440$. Here all the 36 divisors are written in the increasing order. They can be clubbed into 18 pairs, as shown below. (1.1440). (2.720).(3.480) -----etc. ∴ Product of divisors $=(1440{)}^{18}=2^{90}.3^{36}.5^{18}={\left(2^3.3\right)}^{30}.3^6.5^{18}=24^{30}.3^6.5^{18}$ which is divisible by $24{}^x$

∴ Maximum value of $\mathrm{x}=30$