Problem 14 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If the fractional part of the number \(\frac{2^{403}}{15}\) is \(\frac{k}{15}\), then \(\mathrm{k}\). is equal to: \(\quad\) (a) 6 (b) 8 (c) 4 (d) 14

Step-by-Step Solution

Answer

The value of \(k\) is 8, option (b).

Step 1: Understand the Problem

We are given a number \(\frac{2^{403}}{15}\) and need to find the fractional part of it, which is expressed as \(\frac{k}{15}\). The problem requires us to determine the value of \(k\) from the given options.

Step 2: Break Down the Fractional Part

The fractional part of a number \(x\) is the part after the decimal point, which can be expressed as \(x - \lfloor x \rfloor\), where \(\lfloor x \rfloor\) denotes the floor of \(x\), the greatest integer less than or equal to \(x\). For \(\frac{2^{403}}{15}\), it is the difference between the number itself and its integer part.

Step 3: Apply Modulo for the Fraction

The fractional part \(\frac{k}{15}\) can also be determined by finding \(2^{403} \mod 15\). Since \(k\) must satisfy \(0 \leq k < 15\) and represent the remainder, we need to calculate the power mod 15.

Step 4: Use Euler's Theorem to Simplify

Euler's theorem states that if \(a\) and \(n\) are coprime, \(a^{\phi(n)} \equiv 1 \mod n\). Here, \(\phi(15) = 8\) because \(15\) has the prime factors 3 and 5. Therefore, since \(2\) and \(15\) are coprime, \(2^8 \equiv 1 \mod 15\).

Step 5: Calculate the Remainder Using Cycles

Since \(2^8 \equiv 1 \mod 15\), the powers of 2 repeat every 8th exponent. We look for \(2^{403} = 2^{8 \times 50 + 3} = (2^8)^{50} \times 2^3 \equiv 1^{50} \times 2^3 \equiv 2^3 \mod 15\).

Step 6: Calculate \(2^3 \mod 15\)

Calculate \(2^3 = 8\). Hence, \(2^3 \mod 15 = 8\). Therefore, \(2^{403} \mod 15 = 8\).

Step 7: Write the Final Answer

The fractional part \(\frac{k}{15}\) has \(k = 8\), as the remainder of dividing \(2^{403}\) by 15 is \(8\).

Key Concepts

Fractional Part

When dealing with numbers, particularly fractions, a concept that often comes up is the fractional part. The fractional part of a number refers to the portion of it that exists after the decimal point. It's notated mathematically as the difference between the number itself and its integer part, which is the greatest integer less than or equal to the original number.
For example, consider the number 3.76. The integer part is 3, and therefore, the fractional part is 3.76 - 3 = 0.76.

• It is always a non-negative value.
• It should be less than 1 for any number having a fractional part.

In the context of the problem given, we are looking at \(\frac{2^{403}}{15}\) and need to extract it's fractional part in the form of \(\frac{k}{15}\). This entails isolating the remainder when dividing by the denominator, which is dependent on understanding how large powers like \(2^{403}\) behave with respect to a divisor like 15.

Euler's Theorem

Euler's theorem is an important tool in number theory that helps in simplifying the calculation of large powers modulo a number. Specifically, it gives us a shortcut to find remainders quickly. The theorem states that if two numbers, \(a\) and \(n\), are coprime (meaning they have no common factors other than 1), then \(a^{\phi(n)} \equiv 1 \mod n\), where \(\phi(n)\) is Euler's totient function.
Euler's totient function, \(\phi(n)\), gives the count of numbers less than \(n\) that are coprime with \(n\). To find \(\phi(15)\), we note that 15's prime factors are 3 and 5. Thus, \(\phi(15) = 15 \times \left(1 - \frac{1}{3}\right) \times \left(1 - \frac{1}{5}\right) = 8\).
In the exercise, since \(2\) and \(15\) are coprime, Euler's theorem helps us conclude that \(2^8 \equiv 1 \mod 15\), simplifying calculations for large exponents by recognizing the pattern that arises every 8 powers.

Modulo Operation

The modulo operation is a fundamental concept in arithmetic that deals with division and finding remainders. When you perform a modulo operation (often abbreviated as 'mod'), you divide one number by another, and instead of focusing on the quotient, you pay attention to the remainder. For example, dividing 17 by 5 gives a remainder of 2, thus \(17 \mod 5 = 2\).
Knowing how to work with modulus helps in determining remainders efficiently, especially in problems involving powers and large numbers.

• The basic property is: If \(a = b \cdot q + r\), then \(a \mod b = r\), where \(r\) is the remainder.
• In the context of the problem, finding \(2^{403} \mod 15\) revealed the remainder, allowing us to determine that the fractional part \(k\) is 8 since \(2^{403} \equiv 8 \mod 15\).