Quants Questions for CLAT | QB Set 13

A coaching institute conducted a mathematical coding challenge for students preparing for competitive examinations. The challenge focused on the concept of remainders and divisibility. Students were asked to analyse how numbers behave when divided by different integers. One section of the challenge involved arranging books into equal piles. When the books were arranged in groups of 5, 2 books were left over. When arranged in groups of 7, 4 books remained. Another section involved large numbers and powers. Students had to determine the remainder when numbers like 2^10 or 7^15 were divided by smaller integers. The institute explained that remainder-based questions are important because they help in identifying patterns, simplifying calculations, and solving complex arithmetic problems quickly. The instructor also demonstrated how cyclicity works in powers of numbers. Students realised that instead of calculating very large values directly, observing repeating remainder patterns made the questions much easier. By the end of the session, many students became confident in solving divisibility and modular arithmetic problems efficiently.

Questions

Q1. What is the remainder when 2^10 is divided by 7?

A. 1

B. 2

C. 3

D. 2

Q2. A number leaves remainder 2 when divided by 5 and remainder 4 when divided by 7. Which of the following numbers satisfies the condition?

A. 32

B. 27

C. 22

D. 37

Q3. What is the remainder when 7^15 is divided by 5?

A. 1

B. 3

C. 2

D. 4

Q4. What is the remainder when 15^8 is divided by 6?

A. 5

B. 2

C. 3

D. 1

Q5. When a number is divided by 9, the remainder is 5. What will be the remainder when twice the number is divided by 9?

A. 8

B. 2

C. 1

D. 1

Answers with Detailed Explanations

Q1. Correct Answer: D

The powers of 2 modulo 7 follow the cycle 2, 4, 1. Since 10 divided by 3 leaves remainder 1, the corresponding remainder is 2.

Q2. Correct Answer: A

32 leaves remainder 2 when divided by 5 and remainder 4 when divided by 7, satisfying both conditions.

Q3. Correct Answer: B

7 is equivalent to 2 modulo 5. Powers of 2 modulo 5 follow the cycle 2, 4, 3, 1. Since 15 divided by 4 leaves remainder 3, the remainder is 3.

Q4. Correct Answer: C

15 leaves remainder 3 when divided by 6. Any power of 3 divided by 6 continues to leave remainder 3.

Q5. Correct Answer: D

If a number leaves remainder 5 when divided by 9, then twice the number leaves remainder 10. Since 10 divided by 9 leaves remainder 1, the required remainder is 1.