First slide

Permutations

Question

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

Moderate

A

30
B

31
C

32
D

None of these

Solution

In terms of prime factors 33! can be written as 2^a ⋅ 3^b 5^c ⋅ 7d …
Now, E₂ (33!)
$\begin{array}{l} = [\frac{33}{2}] + [\frac{33}{2^{2}}] + [\frac{33}{2^{3}}] + [\frac{33}{2^{4}}] + [\frac{33}{2^{5}}] \\ = 16 + 8 + 4 + 2 + 1 = 31 . \end{array}$
Hence, the exponent of 2 in 33! is 31.
∴ Largest value of n is 31.

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