Si P = 88 × 7n And 96 Divisors Unveiling The Value Of N
In the captivating realm of number theory, we often encounter intriguing problems that challenge our understanding of divisors, prime factorization, and their intricate relationships. Today, we embark on a journey to unravel one such problem: If P = 88 × 7n has 96 divisors, what is the value of 'n'? This problem beckons us to delve into the heart of number theory, where we'll employ the fundamental principles of prime factorization and divisor counting to decipher the enigmatic value of 'n'.
Prime Factorization The Cornerstone of Divisor Analysis
To effectively tackle this problem, our initial step involves prime factorization. Prime factorization is the art of decomposing a composite number into its prime number building blocks. Prime numbers, those indivisible by any number other than 1 and themselves, form the bedrock of number theory. The prime factorization of a number provides a unique fingerprint, revealing the composition of its divisors.
Let's embark on the prime factorization of 88. We can express 88 as 8 × 11. Further decomposing 8, we arrive at 2 × 2 × 2, which can be elegantly written as 2³. Therefore, the prime factorization of 88 is 2³ × 11.
Now, let's incorporate the term 7n into our prime factorization. Since 7 is a prime number, the prime factorization of P can be expressed as 2³ × 11 × 7n. This expression encapsulates the essence of P, laying bare its prime constituents and their respective powers.
Counting Divisors The Art of Enumeration
With the prime factorization of P in hand, we now venture into the realm of divisor counting. The number of divisors of a number is intimately linked to the exponents in its prime factorization. To determine the number of divisors, we employ a simple yet powerful rule: add 1 to each exponent in the prime factorization and then multiply the resulting values.
In our case, the prime factorization of P is 2³ × 11 × 7n. The exponents are 3, 1 (since 11 can be written as 11¹), and n. Applying the rule, we add 1 to each exponent, obtaining 4, 2, and n + 1. Multiplying these values, we arrive at the total number of divisors of P: 4 × 2 × (n + 1) = 8(n + 1).
The Equation Unveiling the Value of n
We are given that P has 96 divisors. Equating our expression for the number of divisors to 96, we obtain the equation 8(n + 1) = 96. This equation holds the key to unlocking the value of 'n'.
Dividing both sides of the equation by 8, we simplify it to n + 1 = 12. Subtracting 1 from both sides, we arrive at the solution: n = 11. Thus, the value of 'n' that satisfies the condition of P having 96 divisors is 11.
Verifying Our Solution Ensuring Accuracy
To ensure the accuracy of our solution, let's substitute n = 11 back into the prime factorization of P: 2³ × 11 × 7¹¹. Now, let's calculate the number of divisors using our rule: (3 + 1) × (1 + 1) × (11 + 1) = 4 × 2 × 12 = 96. This confirms that our solution, n = 11, indeed yields 96 divisors for P.
Conclusion The Power of Prime Factorization
In this enthralling exploration of number theory, we have successfully decoded the value of 'n' in the expression P = 88 × 7n, given that P has 96 divisors. Our journey began with the fundamental concept of prime factorization, which allowed us to dissect P into its prime number constituents. We then harnessed the power of divisor counting, a technique that intimately links the exponents in the prime factorization to the total number of divisors.
By carefully applying these principles, we formulated an equation and solved for 'n', ultimately arriving at the solution n = 11. Our verification step further solidified our confidence in the accuracy of our result. This problem serves as a testament to the elegance and power of number theory, showcasing how seemingly complex problems can be unravelled through the application of fundamental concepts.
Let's delve into a fascinating problem in number theory where we are given that P = 88 × 7n has 96 divisors. Our goal is to determine the value of n. This problem combines the concepts of prime factorization, divisor counting, and algebraic manipulation. To successfully solve it, we need to break down the given information and apply the relevant mathematical principles systematically.
The Importance of Prime Factorization
Our initial step towards solving this problem involves understanding prime factorization. Prime factorization is a crucial technique in number theory, as it allows us to express any composite number as a unique product of prime numbers raised to certain powers. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The prime factorization of a number helps us to identify its divisors and understand its composition in terms of prime numbers.
To begin, let's find the prime factorization of 88. We can express 88 as a product of its prime factors: 88 = 8 × 11. Further, we can decompose 8 into powers of 2: 8 = 2 × 2 × 2 = 2³. Thus, the prime factorization of 88 is 2³ × 11. This representation is fundamental as it simplifies further calculations and helps in understanding the structure of the number.
Now, let’s incorporate the term 7n into our analysis. Since 7 is a prime number, the complete prime factorization of P can be written as: P = 2³ × 11¹ × 7n. Here, we explicitly show the power of 1 for 11 to emphasize the structure of the prime factorization. This expression is key because it forms the basis for counting the number of divisors of P.
Counting the Divisors
With the prime factorization of P established, the next step is to calculate the number of divisors. The number of divisors of a number can be determined directly from its prime factorization. If a number N can be expressed in its prime factorized form as N = p₁ᵃ¹ × p₂ᵃ² × ... × pkᵃ⁋, where p₁, p₂, ..., pk are distinct prime numbers and a₁, a₂, ..., ak are their respective exponents, then the number of divisors d(N) can be calculated using the formula:
d(N) = (a₁ + 1) × (a₂ + 1) × ... × (ak + 1)
This formula stems from the fact that any divisor of N can be formed by selecting a power of each prime factor pi between 0 and ai, inclusive. There are ai + 1 choices for the exponent of each prime factor, and the product of these choices gives the total number of divisors.
Applying this principle to our problem, the prime factorization of P is 2³ × 11¹ × 7n. The exponents are 3, 1, and n. Using the formula for the number of divisors, we get:
Number of divisors of P = (3 + 1) × (1 + 1) × (n + 1) = 4 × 2 × (n + 1) = 8(n + 1)
This algebraic expression represents the total number of divisors of P in terms of n. The next step is to use the given information that P has 96 divisors.
Formulating and Solving the Equation
We are given that P has 96 divisors. Therefore, we can set the expression we derived for the number of divisors of P equal to 96. This gives us the equation:
8(n + 1) = 96
To solve for n, we need to perform algebraic manipulations to isolate n on one side of the equation. First, divide both sides of the equation by 8:
n + 1 = 96 / 8
n + 1 = 12
Next, subtract 1 from both sides of the equation:
n = 12 - 1
n = 11
Thus, the value of n that satisfies the given condition is 11. This is our solution to the problem.
Verification and Conclusion
To ensure our solution is correct, we can verify it by substituting n = 11 back into the expression for the number of divisors of P and checking if it equals 96.
If n = 11, the prime factorization of P becomes 2³ × 11¹ × 7¹¹. The number of divisors of P is then:
(3 + 1) × (1 + 1) × (11 + 1) = 4 × 2 × 12 = 96
This confirms that when n = 11, P indeed has 96 divisors, which validates our solution.
In conclusion, by employing the principles of prime factorization and the formula for counting divisors, we have successfully determined that the value of n is 11. This problem demonstrates the importance of understanding number theory concepts and applying them systematically to solve mathematical challenges. The combination of prime factorization, divisor counting, and algebraic manipulation is a powerful tool in solving problems of this nature.
Let's tackle a compelling problem from number theory that involves prime factorization and divisor counting. The problem states: Given Si P = 88 × 7n has 96 divisors, find the value of "n". This problem is a beautiful blend of number theory and algebra, requiring us to decompose a number into its prime factors and apply the divisor counting formula. We will explore the steps to solve this problem systematically, emphasizing the underlying mathematical principles.
Understanding Prime Factorization
At the heart of this problem lies the concept of prime factorization. Prime factorization is the process of expressing a composite number as a product of its prime factors. A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. For instance, 2, 3, 5, 7, 11, and 13 are prime numbers. Every composite number can be uniquely expressed as a product of prime numbers raised to certain powers. This unique representation is known as the Fundamental Theorem of Arithmetic, which forms the cornerstone of many number-theoretic concepts.
The first step in solving our problem is to find the prime factorization of 88. We can write 88 as a product of 8 and 11: 88 = 8 × 11. Now, we decompose 8 further into its prime factors. Since 8 is 2 cubed, we have 8 = 2³. Therefore, the prime factorization of 88 is 2³ × 11. This decomposition is crucial because it allows us to analyze the structure of P in terms of its prime factors.
Next, we incorporate the term 7n into the prime factorization of P. Since 7 is a prime number, we can write P in its prime factorized form as:
P = 2³ × 11¹ × 7n
Here, we have expressed P as a product of distinct prime factors (2, 11, and 7) raised to their respective powers. This representation is essential for counting the number of divisors of P.
The Divisor Counting Formula
Now that we have the prime factorization of P, we turn our attention to counting its divisors. The number of divisors of a positive integer can be determined directly from its prime factorization. If a positive integer N can be written in its prime factorized form as:
N = p₁ᵃ¹ × p₂ᵃ² × ... × pkᵃ⁋
where p₁, p₂, ..., pk are distinct prime numbers and a₁, a₂, ..., ak are non-negative integers, then the number of divisors of N, denoted as d(N), is given by the formula:
d(N) = (a₁ + 1) × (a₂ + 1) × ... × (ak + 1)
This formula arises from the fact that any divisor of N is of the form p₁ᵇ¹ × p₂ᵇ² × ... × pkᵇ⁋, where each bi is an integer such that 0 ≤ bi ≤ ai. For each prime factor pi, there are ai + 1 choices for the exponent bi. Thus, the total number of divisors is the product of the number of choices for each prime factor.
Applying this formula to our problem, we know that the prime factorization of P is 2³ × 11¹ × 7n. The exponents are 3, 1, and n. Therefore, the number of divisors of P is:
d(P) = (3 + 1) × (1 + 1) × (n + 1) = 4 × 2 × (n + 1) = 8(n + 1)
This expression gives us the number of divisors of P in terms of n. We are given that P has 96 divisors, so we can set this expression equal to 96 and solve for n.
Solving for n
We are given that P has 96 divisors. Therefore, we can set the expression for the number of divisors of P equal to 96:
8(n + 1) = 96
To solve for n, we first divide both sides of the equation by 8:
n + 1 = 96 / 8
n + 1 = 12
Now, we subtract 1 from both sides of the equation:
n = 12 - 1
n = 11
Thus, the value of n that satisfies the given condition is 11. This is our solution to the problem.
Verification
To ensure our solution is correct, we can substitute n = 11 back into the expression for the number of divisors of P and check if it equals 96. When n = 11, the prime factorization of P is 2³ × 11¹ × 7¹¹. The number of divisors of P is then:
(3 + 1) × (1 + 1) × (11 + 1) = 4 × 2 × 12 = 96
This confirms that when n = 11, P indeed has 96 divisors, which validates our solution.
Conclusion
In conclusion, by employing the principles of prime factorization and the divisor counting formula, we have successfully determined that the value of n is 11. This problem highlights the power of number theory and its applications in solving intricate mathematical problems. The combination of prime factorization, divisor counting, and algebraic manipulation provides a systematic approach to finding the unknown value of n. This exercise also reinforces the importance of the Fundamental Theorem of Arithmetic in number theory and its role in understanding the structure of integers and their divisors.