Solution
Given:• F is a number whose value is equal to the product of all the numbers from 2 to 89.
o Hence, F is equal to 89!
• Also, n is the greatest integer, such that, \(12^n\) is a factor of F.
o In other words, n is the highest power of 12 that can divide F.
To find:• The value of the greatest integer n, for which \(12^n\) is a factor of F.
Approach and Working: As 12 is a composite number, first we need to prime factorize it in terms of the prime factors and their corresponding powers.
Next, we need to find the individual instances of 2 and 3.
• Instances of 2: \(\frac{89}{2^1} + \frac{89}{2^2} + \frac{89}{2^3} + \frac{89}{2^4} + \frac{89}{2^5} + \frac{89}{2^6} = \frac{89}{2} + \frac{89}{4} + \frac{89}{8} + \frac{89}{16} + \frac{89}{32} + \frac{89}{64} = 44 + 22 + 11 + 5 + 2 + 1 = 85\)
• Similarly, instances of 3: \(\frac{89}{3^1} + \frac{89}{3^2} + \frac{89}{3^3} + \frac{89}{3^4} = \frac{89}{3} + \frac{89}{9} + \frac{89}{27} + \frac{89}{81} = 29 + 9 + 3 + 1 = 42\)
Now, the number 12 is formed by two instances of 2 and one instance of 3.
• From 85 instances of 2's, number of \(2^2\) can be formed \(= \frac{85}{2} = 42\)
• Hence, number of pairs possible of \(2^2\) and 3 = minimum (42, 42) = 42
Hence, the correct answer is option D.
Answer: D