Given That X And Y Are Natural Numbers, And 380! = 100^x * Y, What Is The Largest Possible Value Of X?

Jun 19, 2025 by ADMIN 103 views

Unveiling the Maximum Power of x in 380! = 100^x * y: A Comprehensive Mathematical Exploration

#h1 Unveiling the Maximum Power of x in 380! = 100^x * y

In the fascinating realm of number theory, problems involving factorials and prime factorization often present intriguing challenges. This article delves into a specific problem: given that x and y are natural numbers and 380! = 100^x * y, our objective is to determine the maximum possible value of x. This exploration will require us to understand the fundamental concepts of factorials, prime factorization, and how to effectively count the occurrences of prime factors within a factorial. Let's embark on this mathematical journey, unraveling the intricacies of this problem and arriving at the solution.

Understanding the Problem: Deconstructing 380! = 100^x * y

To effectively tackle this problem, we must first grasp the core concepts involved. The notation 380! (380 factorial) represents the product of all positive integers from 1 to 380. This massive number contains a multitude of prime factors, each appearing a certain number of times. The equation 380! = 100^x * y essentially states that 380! can be expressed as a product of 100 raised to some power x, and another natural number y. Our goal is to find the largest possible value for x.

Since 100 can be factored into 2² * 5², the problem boils down to finding the highest power of 100 (or equivalently, the highest powers of 2 and 5) that divides 380!. The exponent x will be limited by the prime factor that appears less frequently in the prime factorization of 380!. This is because we need both a 2² and a 5² to form 100. Therefore, we need to determine the number of times 2 and 5 appear as prime factors in 380!.

To achieve this, we will employ Legendre's Formula, a powerful tool for calculating the exponent of a prime p in the prime factorization of n!. This formula allows us to efficiently determine the number of times a prime factor appears in a factorial, which is crucial for solving our problem. By applying Legendre's Formula to the prime factors 2 and 5, we can identify the limiting factor and subsequently determine the maximum value of x.

Legendre's Formula: A Key to Unlocking Factorial Prime Factorization

Legendre's Formula provides a systematic way to determine the exponent of a prime number p in the prime factorization of n!. The formula is expressed as follows:

v_p(n!) = ∑_i=1^∞ ⌊n / pⁱ⌋

where:

v_p(n!) represents the exponent of the prime p in the prime factorization of n!
⌊x⌋ denotes the floor function, which gives the largest integer less than or equal to x.
The summation continues until pⁱ becomes greater than n, at which point the terms become zero.

In simpler terms, Legendre's Formula involves summing the quotients obtained by successively dividing n by increasing powers of p and taking the floor of each quotient. This process effectively counts the number of multiples of p, p², p³, and so on, that are less than or equal to n. Each multiple contributes a factor of p to the factorial, and the sum of these contributions gives the total exponent of p in n!.

This formula is crucial for our problem because it allows us to efficiently calculate the number of times the prime factors 2 and 5 appear in 380!. Without Legendre's Formula, we would have to manually count the multiples of 2 and 5, which would be a tedious and time-consuming process. By applying Legendre's Formula, we can quickly determine the exponents of 2 and 5 in 380!, enabling us to find the maximum value of x.

Applying Legendre's Formula to 380! for Prime Factors 2 and 5

Now, let's apply Legendre's Formula to determine the exponents of the prime factors 2 and 5 in 380!.

Exponent of 2 in 380! (v₂(380!))

v₂(380!) = ⌊380 / 2⌋ + ⌊380 / 4⌋ + ⌊380 / 8⌋ + ⌊380 / 16⌋ + ⌊380 / 32⌋ + ⌊380 / 64⌋ + ⌊380 / 128⌋ + ⌊380 / 256⌋

v₂(380!) = 190 + 95 + 47 + 23 + 11 + 5 + 2 + 1 = 374

This calculation reveals that the prime factor 2 appears 374 times in the prime factorization of 380!.

Exponent of 5 in 380! (v₅(380!))

v₅(380!) = ⌊380 / 5⌋ + ⌊380 / 25⌋ + ⌊380 / 125⌋

v₅(380!) = 76 + 15 + 3 = 94

This calculation shows that the prime factor 5 appears 94 times in the prime factorization of 380!.

These results are crucial for determining the maximum value of x. Since 100 = 2² * 5², we need pairs of 2² and 5² to form 100. The number of times we can form 100 is limited by the prime factor that appears less frequently. In this case, 5 appears 94 times, while 2 appears 374 times. Therefore, the exponent of 5 will be the limiting factor.

Determining the Maximum Value of x: The Limiting Factor

We have established that the prime factor 2 appears 374 times and the prime factor 5 appears 94 times in the prime factorization of 380!. Since 100 = 2² * 5², we need two factors of 2 and two factors of 5 to create one factor of 100. To determine the maximum value of x in the equation 380! = 100^x * y, we need to consider the limiting factor between the number of 2² pairs and the number of 5² pairs.

We can form 374 / 2 = 187 pairs of 2². Similarly, we can form 94 / 2 = 47 pairs of 5². The limiting factor is the smaller of these two values, which is 47. This means that we can form a maximum of 47 factors of 100 from the prime factorization of 380!.

Therefore, the maximum value of x in the equation 380! = 100^x * y is 47. This is because we can express 380! as 100⁴⁷ multiplied by some other natural number y. Any value of x greater than 47 would require more factors of 2 and 5 than are available in the prime factorization of 380!.

This result highlights the importance of prime factorization in understanding the divisibility properties of factorials. By carefully analyzing the exponents of prime factors, we can solve problems that might initially seem complex and daunting. In this case, Legendre's Formula provided a powerful tool for efficiently determining the exponents of 2 and 5, leading us to the solution.

Conclusion: The Maximum Value of x is 47

In conclusion, by leveraging Legendre's Formula and understanding the principles of prime factorization, we have successfully determined the maximum value of x in the equation 380! = 100^x * y. The exponent of the prime factor 5, which appeared 94 times in the prime factorization of 380!, proved to be the limiting factor. After dividing the exponent of 5 by 2 (since we need pairs of 5² to form 100), we found that the maximum value of x is 47.

This problem exemplifies the beauty and power of number theory, demonstrating how fundamental concepts can be applied to solve intricate problems. The combination of Legendre's Formula and careful analysis of prime factors allowed us to efficiently arrive at the solution. This exploration serves as a testament to the elegance and depth of mathematical reasoning.