Question

Find the maximum value of n such that 157! is perfectly divisible by ${18}^n$.

• A. 37
• B. 38
• C. 39
• D. 40

Answer 1

The correct option is A 37

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.
Number of 3's in 157! = 52 + 17 + 5 +1 = 75.
Hence, the number of $3^{2}s$ would be $\left[\frac{75}{2}\right]=37$
The answer would be given by the lower of these values. Hence, 37.
Hence, option (a) is correct.