Problem 2 - A Complete Resource Book in Mathematics for JEE Main

Question

For \(n \in N, \frac{n^{5}}{5}+\frac{n^{3}}{3}+\frac{7}{15} n\) is (A) an integer (B) a natural number (C) a positive fraction (D) None of these

Step-by-Step Solution

Answer

The expression is an integer for multiples of 15.

Step 1: Understand the Expression

The given expression is \( \frac{n^{5}}{5}+\frac{n^{3}}{3}+\frac{7}{15} n \). We need to determine the nature of this expression for natural numbers \( n \).

Step 2: Analyze the Components for Integers

Look at each term in the expression separately. \( \frac{n^5}{5} \) is an integer if \( n^5 \) is divisible by 5, \( \frac{n^3}{3} \) is an integer if \( n^3 \) is divisible by 3, and \( \frac{7}{15}n \) is an integer if \( n \) is divisible by 15.

Step 3: Check Divisibility

For \( n^5 \) to be divisible by 5, \( n \) must be a multiple of 5, and for \( n^3 \) to be divisible by 3, \( n \) must be a multiple of 3. To satisfy all terms being integers simultaneously, \( n \) must be a multiple of 15.

Step 4: Verify Divisibility for \( n=15 \)

Plug \( n = 15 \) into the expression: \( \frac{15^5}{5} + \frac{15^3}{3} + \frac{7}{15} \, 15 \). Simplifying, we find each term results in an integer, thus the entire expression is an integer.

Step 5: Generalization

Since \( n=15 \) results in the expression being an integer and given the structure, for any multiple of 15, the expression yields an integer by similar reasoning. Thus, for \( n \in N \), the expression is an integer whenever \( n \) is a multiple of 15.

Key Concepts

Divisibility Rules

Divisibility rules are useful shortcuts that help us determine if a number can be evenly divided by another without doing intricate calculations. These rules apply to different divisors like 2, 3, 5, and so forth.

**Key Points about Divisibility Rules:**

• For a number to be divisible by 2, it should be even, meaning its last digit should be 0, 2, 4, 6, or 8.
• A number is divisible by 3 if the sum of its digits is divisible by 3. For example, 123 is divisible by 3 because 1 + 2 + 3 = 6 is divisible by 3.
• A number is divisible by 5 if it ends in 0 or 5.
• For 10, the number must end in 0.
• More importantly, when dealing with expressions, to check if an entire expression results in an integer, each term involved needs to be checked for divisibility by its denominator.
• In the exercise, the terms involve divisors like 5, 3, and 15, meaning the number itself should be checked against these rules.

Understanding these basic divisibility rules can simplify complex expressions, as seen in the provided math problem, ensuring we can decide if the expression resolves to an integer.

Integers

Integers are numbers that do not have fractional or decimal parts. They include positive numbers, negative numbers, and zero. When working with integers, we often refer to the familiar set: {..., -3, -2, -1, 0, 1, 2, 3, ...}.

**Characteristics of Integers:**

• Integers are whole numbers, so they do not include fractions or decimals.
• Integers can be positive, negative, or zero.
• When evaluating mathematical expressions, an integer result implies each term and any operation will yield a whole number.

In our exercise, we are asked to determine if the entire expression results in an integer. This means each component of the expression must divide cleanly with no remainder. Therefore, knowledge of integers assures that we find complete, indivisible outcomes in each part of the formula.

Natural Numbers

Natural numbers are the numbers we naturally use to count objects. This set includes all positive integers and starts from 1. Natural numbers are represented by the set {1, 2, 3, 4, ...}.

**Understanding Natural Numbers:**

• Natural numbers are positive and non-zero.
• They are used most often in calculations that involve quantities, sequences, and natural sequences.
• In many contexts, natural numbers imply the exclusion of 0, focusing strictly on counting numbers.

Natural numbers are foundational in the provided exercise since the variable \( n \) must be a natural number. When solving or interpreting mathematical expressions like the one in the exercise, it's crucial to understand the range and nature of the inputs, as natural numbers don't include 0 or negative numbers.