Problem 2 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The fractional part of \(\frac{2^{4 n}}{15}\) is (A) \(\frac{2}{15}\) (B) \(\frac{1}{15}\) (C) \(\frac{4}{15}\) (D) none of these

Step-by-Step Solution

Answer

The fractional part is \( \frac{1}{15} \), so the answer is (B).

Step 1: Understand the Fractional Part

The fractional part of a number is defined as the part of the number that remains after subtracting the integer part. For any real number \( x \), the fractional part can be expressed as \( x - \lfloor x \rfloor \), where \( \lfloor x \rfloor \) denotes the floor of \( x \), or the greatest integer less than or equal to \( x \).

Step 2: Express the Problem

We need to find the fractional part of \( \frac{2^{4n}}{15} \). This means we're interested in \( \frac{2^{4n}}{15} - \lfloor \frac{2^{4n}}{15} \rfloor \).

Step 3: Calculate the Integer Part

The integer part of \( \frac{2^{4n}}{15} \) is given by \( \lfloor \frac{2^{4n}}{15} \rfloor \). Determine this by exploring the behavior of \( 2^{4n} \mod 15 \). Note that: 2^4 = 16 and 16 \mod 15 = 1. So, 2^{4n} \equiv 1^n \equiv 1 \mod 15. Thus, the integer part is indeed zero.

Step 4: Calculate the Fractional Part

Since \( \lfloor \frac{2^{4n}}{15} \rfloor = 0 \), the fractional part is simply \( \frac{2^{4n}}{15} \). We have already shown that \( 2^{4n} \equiv 1 \mod 15 \), meaning \( 2^{4n} \) produces a remainder of 1 when divided by 15.

Step 5: Deduce the Fractional Part

From our calculation, \( \frac{2^{4n} - 1}{15} = \frac{1}{15} \). Hence, the fractional part (the remainder divided by 15) is \( \frac{1}{15} \).

Key Concepts

Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers wrap around upon reaching a certain value, known as the modulus. Imagine that you have a clock with numbers from 0 to 14. When you reach the number 15, you start back at 0. In mathematical terms, if we say that one number is congruent to another modulo 15, it means they have the same remainder when divided by 15.
For example, if you compute that 17 modulo 15 equals 2, you mean that when 17 is divided by 15, it leaves a remainder of 2. This property is essential in many areas of mathematics, including number theory and cryptography. It allows for simplifying calculations by reducing numbers to within a specified range using the modulus, in this case, 15.

• For the exercise, modular arithmetic helps simplify the problem of finding the fractional part because it reduces the power expression exponentiated to any multiple of 4, like \(2^{4n}\), to 1.
• This is articulated as \(2^{4n} \equiv 1 \mod 15\), showing that regardless of the integer \(n\), the arithmetic cycle repeats every four multiples, returning a remainder of 1 when divided by 15.

Integer Part

The integer part of a real number is the whole number portion you obtain when chopping off the fractional component. For instance, the integer part of 3.7 is 3, while for -2.9, it is -3, because -3 is the greatest integer less than or equal to -2.9.
In the floor function notation, this is represented by \(\lfloor x \rfloor\). If you can determine the integer part easily, you can also find the fractional part of a number by subtracting the integer part.
In the exercise, understanding the integer part of \(\frac{2^{4n}}{15}\) simplifies the task to finding \(\lfloor \frac{2^{4n}}{15} \rfloor\), which, when explored through modular arithmetic, reveals it to be zero because \(2^{4n} \equiv 1 \mod 15\).

• This deduction means that \(\frac{2^{4n}}{15}\) minus its integer part results in the fractional part discussed in the solution steps.

Remainder Theorem

The remainder theorem is central to our understanding of the connection between polynomial division and remainders. When a polynomial \(f(x)\) is divided by a linear divisor \(x-c\), the remainder is \(f(c)\).
In our modular arithmetic context, however, the concept can be parallel-tracked to understanding remainders in regular integer division. The calculations in the exercise lead to modular equations like \(2^{4n} \equiv 1 \mod 15\), which indicate that a division of \(2^{4n}\) by 15 leaves a remainder that aligns with the value found through the theorem.
So, applying this idea helps deduce that the remainder actually represented the whole functionality within our fraction calculation.

• This insight guides us directly to the fractional part's solution \(\frac{1}{15}\), by focusing on how the modular remainder aids us in deducing fractional parts of divisions over integers.