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

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Re: The value of the number F is equal to the product of all the ......... [#permalink]
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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.
• $$12 = 2^2 * 3^1$$

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.