The value of the number F is equal to the product of all the .........

Solution

• 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.

• The value of the greatest integer n, for which \(12^n\) is a factor of F.

• \(12 = 2^2 * 3^1\)

• 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: \(\frac{89}{3^1} + \frac{89}{3^2} + \frac{89}{3^3} + \frac{89}{3^4} = \frac{89}{3} + \frac{89}{9} + \frac{89}{27} + \frac{89}{81} = 29 + 9 + 3 + 1 = 42\)

• 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