Question

Find the maximum value of n such that 157! is perfectly divisible by ${12}^n$.

• A. 77
• B. 76
• C. 75
• D. 78

Answer 1

The correct option is C 75

No of 2's in $157!=\left[\frac{157}{2}\right]+\left[\frac{157}{4}\right]+\left[\frac{157}{8}\right]\cdots +\left[\frac{157}{128}\right]=$ 78 + 39 + 19 + 9 + 4 + 2 + 1 = 152.
Hence, the number of $2^{2}s$ would be $\left[\frac{152}{2}\right]=76$
Number of 3's in 157! = 52 + 17 + 5 +1 = 75.
The answer would be given by the lower of these values. Hence, 75 (Option c) is correct.