Exploring the Product of All Factors of 52005

Understanding and solving problems related to the product of all factors of a number, especially numbers in the form of 5n, is a fascinating and crucial topic in number theory. Let's delve into the detailed analysis of the problem: What is the product of all factors of 52005? This exploration will not only enhance our understanding of the intrinsic properties of numbers but will also hone our analytical skills.

Identifying the Factors of 52005

The factors of 52005 are quite straightforward. They are of the form 5k, where k ranges from 1 to 2005. Thus, the factors are:

51, 52, 53, ..., 52004, 52005.

Calculating the Product of These Factors

When we need to calculate the product of all factors of a prime number raised to a power, such as 52005, we can use a systematic approach. Let's break down the steps:

Determine the number of factors. For 52005, the number of factors is:

2005 1

which is 2005.

Calculate the product using the sum of the exponents. The product of all factors can be deduced as:

5 1 2 3 ... 2005 2

The sum of the first 2005 natural numbers is given by:

2005 2 2005

Apply the formula:

5 2005(2005 1) 2

Using the formula for the sum of the first n natural numbers, the sum of 1 through 2005 is:

2005 × 2006 2

Thus, the product simplifies to:

5 2005 × 2006 2

Which evaluates to:

5 2011015

Conclusion

The product of all factors of 52005 is thus 52011015.

Keywords

Factors of 5n Product of Factors Sum of AP (Arithmetic Progression)