The largest integer n such that 33! is divisible by 2ⁿ is

In terms of prime factors 33! can be written as 2^a ⋅ 3^b 5^c ⋅ 7d …

Now, E₂ (33!)

$\begin{array}{l}=\left[\frac{33}{2}\right]+\left[\frac{33}{2^2}\right]+\left[\frac{33}{2^3}\right]+\left[\frac{33}{2^4}\right]+\left[\frac{33}{2^5}\right]\\ =16+8+4+2+1=31.\end{array}$

Hence, the exponent of 2 in 33! is 31.

∴ Largest value of n is 31.