Problem 54 - Algebra for JEE ADVANCED

Question

The number of even divisors of the number \(N=12600=2^{3} 3^{2} 5^{2} 7\) is a. 72 b. 54 c. 18 d. none of these

Step-by-Step Solution

Answer

The number of even divisors of 12600 is 54, which corresponds to option b.

Step 1: Factorize the Number

The given number is already factorized: \(N = 12600 = 2^3 \times 3^2 \times 5^2 \times 7^1 \). Here we can see that the number is composed of powers of prime factors.

Step 2: Identify Even Divisors Condition

Even divisors must include at least one '2' from the factor list (since two is the smallest even prime number). So the minimum exponent of '2' in the divisor must be at least 1.

Step 3: Calculate Possible Exponents

For even divisors, the exponent of 2 can be 1, 2, or 3. The exponent of 3 can be 0, 1, or 2, the exponent of 5 can be 0, 1, or 2, and the exponent of 7 can be 0 or 1.

Step 4: Use Formula for Counting Divisors

The formula for counting divisors of N is \((e_1 + 1)(e_2 + 1)...\), where each \(e_i\) is the exponent of a prime factor. For even divisors, using the exponents of 2 to 3 (1 to 3 for exponents of 2 is counted), the formula becomes: \(3 \times (2 + 1) \times (2 + 1) \times (1 + 1)\).

Step 5: Compute the Result

Calculate using the formula: \(3 \times 3 \times 3 \times 2 = 54\). So the number of even divisors of 12600 is 54.

Key Concepts

Prime Factorization

To understand even divisors, we first need to grasp prime factorization. This technique breaks down a number into its prime components. For instance, the number 12600 can be expressed as the product of prime factors: \(2^3 \times 3^2 \times 5^2 \times 7^1\). Here, 2, 3, 5, and 7 are prime numbers, and the exponents (3, 2, 2, and 1) tell us how many times each prime number is a factor in 12600.

• This method simplifies the understanding and calculation of a number's divisors, both even and odd.
• Each factor's exponent can be adjusted to find different combinations of factors that make up the number.

Recognizing prime factorization is crucial in number theory as it paves the way to calculate various properties of integers, such as divisor counting.

Divisor Counting

Divisor counting is a key process in finding how many integers divide a given number without leaving a remainder. To calculate the number of divisors, we use the formula associated with the exponents of its prime factors. If \(N = p_1^{e_1} \times p_2^{e_2} \times \ldots \times p_k^{e_k}\), then the total number of divisors of \(N\) is \((e_1 + 1)(e_2 + 1)...(e_k + 1)\).

• This formula counts all combinations of prime factors, including both odd and even divisors.
• When counting only even divisors, at least one of the divisors must include the factor 2 with an exponent of 1 or more.

Thus, for 12600, even divisors are calculated by considering the possible exponents for 2, while keeping the full range of exponents for other primes.

Exponentiation

Exponentiation is a mathematical operation involving numbers raised to a power. In prime factorization, each prime factor is raised to a specific power, known as an exponent. For example, in \(2^3\), 2 is raised to the power of 3, meaning \(2 \times 2 \times 2 = 8\).

• The exponents indicate the number of times the base (e.g. 2, 3, 5, or 7) is multiplied by itself.
• Adjusting the exponents leads to different divisors within a number.

The flexibility in choosing different exponents enables practitioners to determine various subsets of divisors, like separating even divisors in the exercise.

Number Theory

Number theory is the branch of mathematics focused on the properties and relationships of numbers, particularly integers. Concepts like prime factorization and divisor counting are fundamental tools in this field. Number theory helps to:

• Understand the structure and classification of numbers.
• Explore concepts like divisibility, primality, and factorization.
• Solve practical problems, such as finding specific types of divisors (like even divisors as in our exercise).

A deep dive into number theory can reveal advanced techniques and fascinating aspects of numbers that are not only mathematically significant but also applicable in real-world scenarios, especially in fields like cryptography and algorithm design.