Remainder Theorem Aptitude Questions Explained_ From Basics to Advanced

1. Remainder Theorem Aptitude Questions Explained: From Basics to Advanced The remainder theorem is an essential concept in aptitude tests, particularly in competitive exams and job assessments. Mastering this topic can help you solve division-based problems quickly and efficiently. In this guide, we will explore the remainder theorem from its basics to advanced applications, providing you with tips, tricks, and solved examples to enhance your problem-solving skills. Understanding the Remainder Theorem The remainder theorem states that if a polynomial f(x) is divided by (x - a), the remainder of this division is simply f(a). Although this theorem is primarily used in algebra, its application in aptitude questions revolves around divisibility, number properties, and modular arithmetic. For example, if we have a polynomial f(x) = x^3 - 4x + 2 and we divide it by (x - 2), then according to the remainder theorem, the remainder is f(2) = 2^3 - 4(2) + 2 = 8 - 8 + 2 = 2. Application of the Remainder Theorem in Aptitude Questions In aptitude tests, remainder-related problems typically involve division of numbers rather than polynomials. These problems are solved using: ● Basic division method ● Properties of divisibility ● Concept of modular arithmetic

2. Key Concepts to Solve Remainder Theorem Problems 1. Basic Remainder Calculation When a number N is divided by D, it gives a quotient Q and remainder R, which can be expressed as: N = D × Q + R, where 0 ≤ R < D Example 1: Find the remainder when 25 is divided by 7. Solution: 25 ÷ 7 = 3 remainder 4. Hence, the remainder is 4. 2. Remainder of Large Powers Finding the remainder when large numbers or powers are divided requires modular arithmetic. Example 2: Find the remainder when 2^10 is divided by 3. Solution: Using modular arithmetic: ● 2^1 ≡ 2 (mod 3) ● 2^2 ≡ 4 ≡ 1 (mod 3) ● 2^3 ≡ 2 (mod 3) ● 2^4 ≡ 1 (mod 3) Since the cycle repeats every two steps, 2^10 follows the pattern: 2^10 = (2^2)^5 ≡ 1^5 ≡ 1 (mod 3) So, the remainder is 1. 3. Remainder When Dividing by Multiple Numbers To solve these problems, we use the Chinese Remainder Theorem. Example 3: Find the remainder when 47 is divided by 6 and 5.

3. Solution: ● 47 ÷ 6 → remainder 5 ● 47 ÷ 5 → remainder 2 So, the remainders are 5 and 2 respectively. 4. Using Euler’s Theorem Euler’s theorem states that for any integer a and a prime p, a^(p-1) ≡ 1 (mod p) This helps in solving large exponentiation problems efficiently. Example 4: Find the remainder when 5^100 is divided by 7. Solution: Using Euler’s theorem: Since 5^6 ≡ 1 (mod 7), we break down 5^100 as: 5^100 = (5^6)^16 × 5^4 Since 5^6 ≡ 1, we get: 5^100 ≡ 5^4 (mod 7) 5^4 = 625 625 ÷ 7 → remainder 2. 5. Remainder of Factorial Numbers To find the remainder of factorial numbers, use Wilson’s theorem and properties of modular arithmetic. Example 5: Find the remainder when 10! is divided by 11. Solution: By Wilson’s theorem: (p-1)! ≡ -1 (mod p) For p = 11, we get: 10! ≡ -1 (mod 11) So, the remainder is 10. Tricks and Shortcuts to Solve Remainder Theorem Questions 1. For powers of numbers, look for repeating cycles in modular arithmetic. 2. For large factorials, use Wilson’s theorem when the divisor is a prime number. 3. Break down large numbers using their digit sum or modular properties. 4. Use divisibility rules to simplify remainder calculations quickly.

4. 5. Apply Chinese Remainder Theorem when dividing by multiple numbers. Practice Questions Try solving these remainder theorem questions to test your understanding: 1. Find the remainder when 12345 is divided by 9. 2. Find the remainder when 3^50 is divided by 7. 3. If a number N leaves a remainder 3 when divided by 7 and remainder 5 when divided by 9, find the smallest value of N. 4. Find the remainder when 20! is divided by 23. 5. Compute the remainder when 987654321 is divided by 11. Conclusion Mastering remainder theorem aptitude questions can significantly boost your problem-solving speed in competitive exams. By understanding basic division, modular arithmetic, and advanced concepts like Euler’s and Wilson’s theorems, you can efficiently tackle complex problems. Regular practice and strategic application of these concepts will enhance your aptitude skills and help you ace your exams with confidence. Stay tuned to AptiMentor for more aptitude preparation tips, tricks, and solved examples!