What is the highest power of 54 that can divide 46!?

A. 0
B. 7
C. 12
D. 21
E. 42

Solution

• We are given a factorial value, 46! which is getting divided by a divisor 54.

• The highest power of 54 that can divide 46!

• \(54 = 2^1 * 3^3\)

• Instances of 2: \(\frac{46}{2^1} + \frac{46}{2^2} + \frac{46}{2^3} + \frac{46}{2^4} + \frac{46}{2^5} = \frac{46}{2} + \frac{46}{4} + \frac{46}{8} + \frac{46}{16} + \frac{46}{32} = 23 + 11 + 5 + 2 + 1 = 42\)
• Similarly, instances of 3: \(\frac{46}{3^1} + \frac{46}{3^2} + \frac{46}{3^3} = \frac{46}{3} + \frac{46}{9} + \frac{46}{27} = 15 + 5 + 1 = 21\)

• From 21 instances of 3’s, number of \(3^3\) can be formed \(= \frac{21}{3} = 7\)
• Hence, number of possible pairs of \(2^1\) and \(3^3\) = minimum (42, 7) = 7