Problem 84 - A Complete Resource Book in Mathematics for JEE Main

Question

Number of points having position vector \(a \hat{i}+b \hat{j}+c \hat{k}\) where \(a, b, c \in\\{1,2,3,4,5\\}\) such that \(2^{a}+3^{b}+5^{c}\) is divisible by 4 is (A) 140 (B) 70 (C) 100 (D) None of these

Step-by-Step Solution

Answer

None of these; the correct number is 75.

Step 1: Understanding the Problem

We need to find the number of combinations of \(a, b, c\), each ranging from 1 to 5, such that the expression \(2^a + 3^b + 5^c\) is divisible by 4.

Step 2: Analyze Expression Modulo 4

We will evaluate \(2^a \mod 4\), \(3^b \mod 4\), and \(5^c \mod 4\) for each value of \(a, b,\) and \(c\) from 1 to 5.

Step 3: Compute 2^a Modulo 4

The powers of 2 modulo 4 cycle as follows: \(2^1 \equiv 2\), \(2^2 \equiv 0\), \(2^3 \equiv 0\), \(2^4 \equiv 0\), \(2^5 \equiv 0 \) (mod 4). For \(a = 2, 3, 4, 5\), \(2^a \equiv 0 \mod 4\). Only for \(a = 1,\; 2^a \equiv 2 \mod 4\).

Step 4: Compute 3^b Modulo 4

The powers of 3 mod 4 cycle with period 2: \(3^1 \equiv 3\), \(3^2 \equiv 1\), \(3^3 \equiv 3\), \(3^4 \equiv 1 \) (mod 4). For odd \(b\), \(3^b \equiv 3 \mod 4\); for even \(b\), \(3^b \equiv 1 \mod 4\).

Step 5: Compute 5^c Modulo 4

The powers of 5 modulo 4 are constant: \(5^1 \equiv 1\), \(5^2 \equiv 1\), \(5^3 \equiv 1\), \(5^4 \equiv 1\), \(5^5 \equiv 1 \) (mod 4). Therefore, \(5^c \equiv 1 \mod 4\) for all \(c\).

Step 6: Set Conditions for Divisibility by 4

We require that \(2^a + 3^b + 5^c \equiv 0 \mod 4\). Considering possible values: \(2^a = 2\) only if \(a=1\); \(3^b = 3\) for odd \(b\) and \(1\) for even \(b\); \(5^c = 1\).

Step 7: Calculate Viable Combinations

- If \(a = 1\) (\(2^a \equiv 2\)), then \(3^b + 1 \equiv 2 \mod 4\) requires \(3^b \equiv 3\), so \(b\) must be odd. - For \(b = 1, 3, 5\), \(3^b \equiv 3 \mod 4\) (3 values). - For each valid \(b\), \(c\) can be 1 to 5 (as \(5^c \equiv 1\), no restriction), giving 5 values.Thus, there are 3 (for \(b\)) x 5 (for \(c\)) = 15 combinations when \(a = 1\).

Step 8: Sum All Combinations

For each \(a = 2, 3, 4, 5\), \(2^a \equiv 0\). Therefore, we need \(3^b + 1 \equiv 0 \mod 4\), implying \(3^b \equiv 3 \mod 4\) again, so \(b\) must be odd.- 3 choices for odd \(b\) (1, 3, 5) and 5 choices for \(c\), giving 3 x 5 = 15 combinations.- Corresponding to \(a = 2, 3, 4, 5\), there are 4 \(a\) values x 15 combinations = 60.Thus, total combinations = 60 + 15 = 75.

Key Concepts

Modulo Operations

Modulo operations, often denoted as "mod," involve finding the remainder after division of one number by another. It is a fundamental concept in mathematics, especially useful in operations related to periodicity, cryptography, and computer science.

When working with modulo, the goal is to simplify calculations by focusing only on the remainder. For example, if we say "7 mod 3," we are asking for the remainder when 7 is divided by 3, which is 1.

In the context of the exercise, the expression \(2^a + 3^b + 5^c\) must be divisible by 4, which is the same as saying it must be equivalent to 0 modulo 4. This involves computing \(2^a \mod 4\), \(3^b \mod 4\), and \(5^c \mod 4\) to help determine when their sum is 0 modulo 4.

Understanding different values under modulo helps in predicting patterns and simplifying equations, such as when we say that the powers of 2 after \(2^1\) until \(2^5\) all simplify to 0 mod 4. This makes computations much more straightforward, illustrating the power of radical simplification through modulo operations.

Number Theory

Number theory is a branch of mathematics that is concerned with the properties and relationships of numbers, particularly integers. It covers a wide range of topics such as divisibility, prime numbers, and modular arithmetic.

In this exercise, we see number theory at work through the use of modular arithmetic. Understanding the behavior of powers and their simplifications when divided by a number such as 4 is rooted in number theory. For example, knowing how \(2^a\), \(3^b\), and \(5^c\) behave under modulo 4 allows us to determine combinations of \(a, b,\) and \(c\) that meet the divisibility criteria.

Number theory helps us understand and solve these complex combinations without having to manually check each one. Instead, we rely on established patterns and theorems to simplify these checks. All of this showcases why number theory is considered fundamental in studies of mathematics, with applications from cryptography to computer algorithms.

Exponents and Powers

Exponents and powers in mathematics are ways to represent repeated multiplication of a number by itself. An exponent denotes how many times the base number is multiplied. For instance, \(5^3\) means \(5 \times 5 \times 5\).

In the provided exercise, handling exponents of numbers like 2 and 3 is essential. Knowing how exponents behave when reduced under a specific modulus is crucial. For example, powers of 2 mimic a pattern of \(2, 0, 0, 0\) when considered modulo 4. This shows how mod operations can simplify complex calculations involving powers.

Understanding these concepts allows us to swiftly calculate and deduce that \(2^a\) will be 0 for \(a = 2, 3, 4, 5\), ensuring we select the correct matching conditions for numbers like \(3^b\) and \(5^c\) to satisfy divisibility. Handling exponents correctly is vital as it seeks patterns that ease calculation burdens in problems involving high powers and multiple variables.