Problem 59 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \([x]\) denotes the greatest integer less than or equal to \(x\), then \(\left[(6 \sqrt{6}+14)^{2 n+1}\right]\) (A) is an even integer (B) is an odd integer (C) depends on \(n\) (D) none of these

Step-by-Step Solution

Answer

The expression is an odd integer (B).

Step 1: Understanding the Greatest Integer Function

The greatest integer function, also known as the floor function, denoted as \([x]\), returns the largest integer less than or equal to \(x\). For example, \([3.7] = 3\) and \([5] = 5\).

Step 2: Simplifying the Expression inside the Function

The expression \((6\sqrt{6} + 14)^{2n+1}\) implies repeated multiplication. Notice that if we expand \((6\sqrt{6} + 14)^{2n+1}\), the terms will include powers of 6 and 14, dominated by the integer term as it is larger, but 6√6 is not an integer.

Step 3: Considering Parity of the Expression

Since \((6\sqrt{6} + 14)\) is greater than an integer, its power, \((2n+1)\), is odd. Therefore, \([(6\sqrt{6} + 14)^{2n+1}]\) will have the dominant integer term raised to an odd power, making the entire expression an odd integer.

Step 4: Verifying the Parity

Direct calculations show that if \((2n+1)\) is an odd power, then \(14^{2n+1}\) becomes odd. Adding any non-integer (from the \(6\sqrt{6}\) component) does not change the parity conclusion in the floor function since it rounds down to the nearest integer below. Thus, the result remains odd.

Key Concepts

Parity in Mathematics

Parity in mathematics is a fundamental concept that describes the attribute of integers being either even or odd. Understanding parity can help in predicting the behavior of numbers in various mathematical functions and problems.

Parity works like this:

• An integer is considered even if it is divisible by 2 without a remainder. Examples include numbers like 2, 4, 6, etc.
• An integer is considered odd if it is not divisible by 2, leaving a remainder of 1. Examples include numbers like 1, 3, 5, etc.

Recognizing parity is crucial, especially when dealing with the greatest integer functions and complex expressions involving integers. This can often dictate the outcome of whether a number is even or odd after certain transformations.

Odd and Even Numbers

Odd and even numbers form the foundation of parity in mathematics. Each integer will fall into one of these two categories.

Here's a quick guide:

• An even number can take the form of \(2k\), where \(k\) is an integer.
• An odd number fits the form \(2k + 1\), showcasing that odd numbers are just one more than an even number.

In complex expressions, especially those involving powers, such as the one in the original exercise, identifying whether a factor or base results in an odd or even number is key.

For instance, if you take an even number to any power, it remains even. Conversely, an odd number raised to any power remains odd. This property can be incredibly useful in solving problems where a part of the expression determines the parity.

Exponents and Powers

Exponents and powers play a significant role in transforming numbers and analyzing outcomes in mathematics. Understanding them helps determine parity in complex expressions.

When you deal with exponents:

• The base number is the primary number being raised to a power.
• The exponent tells you how many times to multiply the base by itself.

For example, taking \(14^{2n+1}\) involves raising 14 to an odd power. Since 14 is even, raising it to any power maintains its even property. However, in an expression like \((6\sqrt{6} + 14)^{2n+1}\), the dominant factor with a larger integer part often dictates the parity.

This specific type of analysis is crucial in solving whether a complex expression results in an odd or even integer, especially when wrapped in functions like the greatest integer function.