Problem 68 - A Complete Resource Book in Mathematics for JEE Main

Question

The number of divisors a number 38808 can have, excluding 1 and the number itself is (A) 70 (B) 72 (C) 71 (D) None of these

Step-by-Step Solution

Answer

142 divisors, option (D) "None of these" is correct.

Step 1: Prime Factorization of the Number

To determine the number of divisors, we first need to perform the prime factorization of the number 38808. The prime factorization of 38808 is \[ 38808 = 2^3 \times 3^5 \times 7^2 \times 11^1 \].

Step 2: Calculating Total Number of Divisors

Using the formula for the number of divisors, where for a number given by \( n = p^a \times q^b \times r^c \times \ldots \), the number of divisors is \[(a+1)(b+1)(c+1)\ldots \]. Applying this to 38808, we have:\[ (3+1)(5+1)(2+1)(1+1) = 4 \times 6 \times 3 \times 2 = 144 \]. Thus, 38808 has 144 divisors.

Step 3: Excluding 1 and the Number Itself

Since we need to count the divisors excluding 1 and the number itself, we simply subtract 2 from the total number of divisors. \[ 144 - 2 = 142 \]. Therefore, the number of divisors, excluding 1 and 38808 itself, is 142.

Key Concepts

Prime Factorization

Prime factorization is a fundamental concept in number theory. It involves breaking down a composite number into a product of its prime factors. Primes are numbers greater than one that cannot be formed by multiplying two smaller numbers. The process begins by dividing the original number by the smallest prime number, which is 2, and continues with increasing primes until the remaining part of the number is 1. For instance, consider the number 38808. It can be decomposed into its prime components as:\[ 38808 = 2^3 \times 3^5 \times 7^2 \times 11^1 \]

• 2: Divides evenly into 38808.
• 3: Continues to divide the result.
• 7: Further divides the subsequent quotient.
• 11: Finally breaks down the number into manageable factors.

The process of prime factorization is crucial as it provides a unique representation of the number and serves as the foundation for various number theoretical techniques such as finding divisors.

Divisors Calculation

Calculating the number of divisors of a number using its prime factorization is an essential part of problem-solving in mathematics. Once a number has been expressed as the product of its primes, say \( n = p^a \times q^b \times r^c \cdots \), the total number of divisors is computed with the formula:\[ (a+1)(b+1)(c+1)\ldots \]For example, using the prime factorization of 38808, \( 38808 = 2^3 \times 3^5 \times 7^2 \times 11^1 \), the number of divisors is calculated as follows:

• Add 1 to each of the exponents: \( (3+1), (5+1), (2+1), (1+1) \).
• Multiply these results together: \( 4 \times 6 \times 3 \times 2 = 144 \).

This process not only gives us the total count of divisors but also sets the stage for more advanced operations, such as excluding specific divisors. Such computations are invaluable in arithmetic, algebra, and beyond.

Mathematical Problem Solving

Mathematical problem solving is a critical skill in understanding concepts like divisor calculation. It involves identifying the problem, planning a strategy, and executing the solution with precision. In the exercise given, after calculating the total number of divisors, the next step is to refine the solution by considering specific conditions, such as excluding certain divisors.
For the number 38808, which has 144 divisors in total, it's often important in problems to exclude certain values, like 1 and 38808 itself. Subtracting these from the total gives: \[144 - 2 = 142 \].
This adjustment in problem-solving ensures accuracy in meeting the problem's requirements. Successful mathematical problem solving combines understanding the basics, applying them to a structured process, and refining the results to meet specific criteria, showcasing the beauty of methodical thinking in mathematics.