Problem 68 - A Complete Resource Book in Mathematics for JEE Main 2017

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

(D) None of these

Step 1: Prime Factorize the Number

Find the prime factors of 38808. First, divide by the smallest prime number, 2, until no longer divisible:\[ 38808 \div 2 = 19404 \19404 \div 2 = 9702 \9702 \div 2 = 4851 \]4851 is not divisible by 2, so try the next prime, 3: \[ 4851 \div 3 = 1617 \1617 \div 3 = 539 \539 \div 11 = 49 \49 \div 7 = 7 \] which leaves 1. Thus, the prime factorization is \( 38808 = 2^3 \times 3^2 \times 7 \times 11 \).

Step 2: Calculate the Total Number of Divisors

Using the formula for calculating the total number of divisors from the prime factorization, if a number \( n \) has the prime factorization \( p_1^{k_1} \times p_2^{k_2} \times \ldots \times p_m^{k_m} \), the number of divisors is \( (k_1 + 1) \times (k_2 + 1) \times \ldots \times (k_m + 1) \).So, for 38808 we calculate: \( (3+1)(2+1)(1+1)(1+1) = 4 \times 3 \times 2 \times 2 = 48 \).Thus, 38808 has 48 divisors.

Step 3: Exclude 1 and the Number Itself

The problem asks for the number of divisors excluding 1 and the number itself. Since 38808 has 48 divisors total, excluding 1 and 38808 leaves:\[ 48 - 2 = 46 \].

Key Concepts

Prime Factorization

Prime factorization is an essential step in breaking down a number into its basic building blocks -- the prime numbers. A prime number is a natural number greater than 1, which has no other divisors than 1 and itself. To perform a prime factorization, you start with the smallest prime number, which is 2.
If the number is divisible by 2, you continue to divide by 2 until it is no longer possible. Then, you proceed to the next smallest prime, which is 3, and continue similarly.

For example, when we prime factorize 38808, we start with:

• 38808 is divisible by 2, so divide until it's no longer possible.
• Next, proceed with 3 for the remaining number.
• Continue with higher primes like 7 and 11 as needed.

The factored result gives us: \[ 38808 = 2^3 \times 3^2 \times 7 \times 11 \] This step is crucial because the prime factors form the foundation upon which other number properties are built. Understanding the prime composition helps in determining other attributes of the number, such as divisibility and the calculation of divisors.

Number Theory

Number theory is a fundamental branch of mathematics exploring the properties and relationships of numbers, particularly integers. It examines integers, prime numbers, and the numerous patterns and problems related to them.
Number theory is intrinsic to solving problems like finding divisors, because it provides the underlying rules and logic.

For example, it tells us:

• The principle of divisibility and how numbers break down into constituents.
• Why every integer has a unique prime factorization.
• How mathematical properties play a role in calculations.

In our exercise, number theory aids in finding all possible divisors through understanding prime factor composition. It allows us to analyze the number 38808 effectively by recognizing its prime components and subsequently using them to determine overall divisibility.

Divisor Formula

The divisor formula is a method derived from the unique prime factorization of a number to determine how many factors it has. Consider a number with a prime factorization of the form \( p_1^{k_1} \times p_2^{k_2} \times \ldots \times p_m^{k_m} \).
The divisor formula states that the total number of divisors can be calculated by the product of each of the exponents incremented by one:
\[(k_1 + 1)(k_2 + 1)\ldots(k_m + 1)\].

Applying this to 38808's factorization:

• From \( 2^3 \times 3^2 \times 7 \times 11 \), calculate \((3 + 1)(2 + 1)(1 + 1)(1 + 1)\).
• This results in \(4 \times 3 \times 2 \times 2 = 48\) total divisors.

To find only the proper divisors, exclude 1 and 38808 itself, leaving \(48 - 2 = 46\).
This formula simplifies a complex process by transforming the prime factorization directly into meaningful information about the number’s divisibility.