Mathematics - Permutation & Combination Question with Solution | TestHub

$1440=2^5\cdot 3^2\cdot 5^1$

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

Product of divisors $=1\cdot 2\cdot 3\dots 180\cdot 720\cdot 1140$. Here, all 36 divisors are written in increasing order. They can be grouped into 18 pairs, as shown below.

$(1\cdot 1440)\cdot (2\cdot 720)\cdot (3\cdot 480)\cdot etc.$

Therefore, the product of divisors $=(1440{)}^{18}=2^{90}\cdot 3^{36}\cdot 5^{18}=(2^3\cdot 3{)}^{30}\cdot 3^6\cdot 5^{18}=24^{30}\cdot 3^6\cdot 5^{18}$, which is divisible by $24^x$.

Therefore, the maximum value of $x=30$.