The largest \(\mathrm{n}\in \mathrm{N}\) such that \({3}^{\mathrm{n}}\) divides \(50\) ! is: [JEE Main 2025, 2 Apr (Shif…

To find the exponent of a prime \(p\) in \(n!\), we use:

\(\text{Exponent of }p\text{ in }n!=\sum _{k=1}^{\mathrm{∞}}⌊\frac{n}{{p}^{k}}⌋\)

Here, \(p=3\), and \(n=50\). So,

\(\sum _{k=1}^{\mathrm{∞}}⌊\frac{50}{{3}^{k}}⌋=⌊\frac{50}{3}⌋+⌊\frac{50}{9}⌋+⌊\frac{50}{27}⌋+⌊\frac{50}{81}⌋+\ldots\)

Now compute:

• \(⌊\frac{50}{3}⌋=16\)
• \(⌊\frac{50}{9}⌋=5\)
• \(⌊\frac{50}{27}⌋=1\)
• \(⌊\frac{50}{81}⌋=0\)

Adding up:

\(16+5+1=22\)