Key Takeaways From the Blog
- Mastering number system aptitude questions is crucial for excelling in quantitative aptitude sections of competitive exams and placements.
- Key topics include types of numbers, HCF/LCM, divisibility, digit manipulation, and word problems.
- Regular practice with diverse question types and formulas boosts speed and accuracy.
- Memorize essential formulas and apply shortcut tricks for efficient problem-solving.
- Use adaptive practice and review detailed solutions to strengthen understanding and exam readiness.
Introduction
A strong foundation in the number system is vital for anyone aiming to excel in quantitative aptitude. The number system forms the backbone of most mathematical reasoning, making it an essential topic for students, job seekers, and professionals alike. Whether you’re preparing for competitive exams, campus placements, or simply want to sharpen your analytical skills, understanding the number system gives you a distinct advantage. It not only helps you tackle a variety of quantitative questions but also builds confidence in approaching complex problems.
This comprehensive guide is designed to help you master number system aptitude questions from the ground up. Inside, you’ll discover key concepts, important formulas, common question types, and proven strategies to boost your speed and accuracy. By following the tips and practice resources provided, you’ll be well-equipped to handle any number system question—no matter the exam or challenge you face.
What is the Number System in Aptitude?
The number system refers to the mathematical structure used to represent, classify, and manipulate numbers. It includes natural numbers, whole numbers, integers, rational and irrational numbers, and real numbers. A clear understanding of these categories is crucial for solving related questions efficiently.
Why Practice Number System Aptitude Questions?
Number system aptitude questions frequently appear in exams like SSC, Banking, GATE, CAT, and TCS placement tests. These questions form the backbone of the quantitative aptitude section and test your grasp of fundamental mathematical principles. Regular practice helps you develop speed, accuracy, and analytical thinking.
Quick Note: Consistent practice with number system questions is a proven way to boost your quantitative aptitude scores.
Key Topics in Number System Aptitude Questions
Mastering the number system involves understanding a variety of key topics. Each topic tests a different aspect of your mathematical reasoning and is crucial for tackling competitive exam questions.
- Types of Numbers: Learn to distinguish between natural numbers, whole numbers, integers, rational and irrational numbers, and real numbers.
- HCF and LCM: Calculating the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) is a frequent theme. Practice using prime factorization and shortcut formulas for efficiency.
- Divisibility Rules and Remainders: These questions often require you to apply divisibility rules or find remainders using modular arithmetic.
- Unit Digit and Last Digit Problems: Finding the unit digit of large exponentiated numbers is a common challenge. Use cyclicity patterns to quickly find answers.
- Sum, Difference, and Manipulation of Digits: Many questions involve operations with digits, such as reversing digits, finding sums or differences, or forming new numbers.
- Prime Numbers and Composite Numbers: Identifying prime numbers, checking for twin primes, and distinguishing between prime and composite numbers are fundamental skills.
- Factorization and Number of Factors: These questions require you to break down numbers into their prime factors, count the total number of factors, or find the sum of all factors.
- Ratio, Percentage, and Age-Based Number Questions: Some questions blend arithmetic with ratios, percentages, or age relationships. These problems require logical deduction and the ability to translate words into equations.
- Word Problems and Applications: Real-life scenarios, such as distributing objects, calculating ages, or solving puzzles involving numbers, are often included.
- Competitive and Placement-Specific Questions: For those preparing for exams like GATE or TCS, you’ll encounter challenging questions that may combine several concepts in one problem.
Quick Note: By focusing your preparation on these key topics, you’ll be ready to tackle any number system questions.
Memorizing key formulas will help you solve number system questions quickly and accurately. Here are some of the most useful formulas:
- Sum of First n Natural Numbers:
Sum = n(n + 1)/2
Example: The sum of the first 20 natural numbers is 20 × 21 / 2 = 210. - Sum of First n Odd Numbers:
Sum = n²
Example: The sum of the first 7 odd numbers is 7² = 49. - Sum of First n Even Numbers:
Sum = n(n + 1)
Example: The sum of the first 6 even numbers is 6 × 7 = 42. - Product of First n Natural Numbers (n factorial):
Product = n! = n × (n – 1) × (n – 2) × … × 1
Example: 5! = 5 × 4 × 3 × 2 × 1 = 120. - LCM and HCF Relationship:
LCM × HCF = Product of the two numbers
Example: If HCF = 3, LCM = 60, numbers could be 12 and 15 (since 12 × 15 = 180 = 3 × 60). - Number of Factors of a Number:
If a number N = a^p × b^q × c^r (where a, b, c are prime factors),
Number of factors = (p + 1)(q + 1)(r + 1) - Sum of Factors of a Number:
If N = a^p × b^q × c^r,
Sum of factors = [(a^(p+1) – 1)/(a – 1)] × [(b^(q+1) – 1)/(b – 1)] × [(c^(r+1) – 1)/(c – 1)] - Divisibility Rules:
- A number is divisible by 2 if its last digit is even.
- Divisible by 3 if the sum of its digits is divisible by 3.
- Divisible by 4 if the last two digits form a number divisible by 4.
- Divisible by 5 if the last digit is 0 or 5.
- Divisible by 6 if divisible by both 2 and 3.
- Divisible by 9 if the sum of its digits is divisible by 9.
- Divisible by 10 if the last digit is 0.
- Unit Digit of a Power:
Find the cyclicity (pattern repeats every few powers), then use the exponent mod cycle length to determine the unit digit. - Sum of Digits of a Number:
Useful for divisibility by 3 and 9.
Bottom Line: Keep these formulas handy as you practice number system aptitude questions—they’ll save you time and help you solve problems with confidence!
Number System Aptitude Tricks
To solve aptitude questions and answers on number system faster, learn number system aptitude tricks such as:
- Recognizing patterns in unit digits and remainders,
- Applying divisibility rules,
- Using shortcut formulas for HCF and LCM,
- Breaking down complex word problems into smaller, manageable parts.
Sample Aptitude Questions for Number System
Here are some sample types of aptitude test number system questions you might encounter:
1. Find the HCF of 36 and 60.
Answer: 12
Explanation:
36 = 2² × 3²
60 = 2² × 3 × 5
HCF = 2² × 3 = 12
2. Find the LCM of 15 and 20.
Answer: 60
Explanation:
15 = 3 × 5
20 = 2² × 5
LCM = 2² × 3 × 5 = 60
3. What is the unit digit of 7^45?
Answer: 3
Explanation:
Unit digit of powers of 7: 7, 9, 3, 1 (repeats every 4).
45 ÷ 4 = 11 remainder 1 → 7
So, unit digit is 7.
4. Which of the following is not a prime number: 2, 3, 5, 9?
Answer: 9
Explanation:
9 = 3 × 3, so it is not prime.
5. What is the sum of the first 10 even numbers?
Answer: 110
Explanation:
Sum = n(n + 1) = 10 × 11 = 110
6. What is the remainder when 2^31 is divided by 5?
Answer: 2
Explanation:
Powers of 2 mod 5 repeat every 4: 2, 4, 3, 1.
31 ÷ 4 = 7 remainder 3 → 3rd term = 3
7. Find the sum of all factors of 18.
Answer: 39
Explanation:
Factors: 1, 2, 3, 6, 9, 18. Sum = 39
8. If the product of two numbers is 120 and their HCF is 6, what is their LCM?
Answer: 20
Explanation:
Product = HCF × LCM → 120 = 6 × LCM → LCM = 20
9. Which of the following are twin primes: (3, 5), (5, 7), (11, 13), (17, 19)?
Answer: All pairs
Explanation:
All pairs differ by 2 and both are primes.
10. What is the smallest natural number?
Answer: 1
Explanation:
Natural numbers start from 1.
11. Find the LCM of 12, 15, and 20.
Answer: 60
Explanation:
12 = 2² × 3, 15 = 3 × 5, 20 = 2² × 5
LCM = 2² × 3 × 5 = 60
12. What is the HCF of 81 and 54?
Answer: 27
Explanation:
81 = 3⁴, 54 = 2 × 3³
HCF = 3³ = 27
13. The sum of two numbers is 45 and their difference is 9. What are the numbers?
Answer: 27 and 18
Explanation:
Let numbers be x and y.
x + y = 45, x - y = 9 → x = 27, y = 18
14. What is the unit digit of 6^99?
Answer: 6
Explanation:
Unit digit of powers of 6 is always 6.
15. Find the number of factors of 36.
Answer: 9
Explanation:
36 = 2² × 3². Number of factors = (2+1) × (2+1) = 9
16. What is the sum of the digits of 473?
Answer: 14
Explanation:
4 + 7 + 3 = 14
17. If 30% of a number is 45, what is the number?
Answer: 150
Explanation:
0.3x = 45 → x = 150
18. Find the smallest odd prime number.
Answer: 3
Explanation:
2 is the smallest prime, but 3 is the smallest odd prime.
19. If the sum of two numbers is 50 and their HCF is 5, what could be the numbers?
Answer: 5 and 45, 15 and 35, 25 and 25
Explanation:
Numbers must be multiples of 5 that sum to 50.
20. What is the remainder when 12345 is divided by 9?
Answer: 6
Explanation:
Sum of digits = 1+2+3+4+5=15; 1+5=6
21. What is the greatest 3-digit number divisible by 7?
Answer: 994
Explanation:
999 ÷ 7 = 142 remainder 5. 142 × 7 = 994
22. What is the least common multiple of 8 and 14?
Answer: 56
Explanation:
8 = 2³, 14 = 2 × 7; LCM = 2³ × 7 = 56
23. Find the HCF of 42 and 56.
Answer: 14
Explanation:
42 = 2 × 3 × 7; 56 = 2³ × 7; HCF = 2 × 7 = 14
24. The difference between two numbers is 36. If their ratio is 5:2, what are the numbers?
Answer: 60 and 24
Explanation:
Let numbers be 5x and 2x. 5x - 2x = 36 → x = 12; 5x = 60, 2x = 24
25. Find the unit digit of 9^37.
Answer: 9
Explanation:
Unit digit of powers of 9: 9, 1, 9, 1… (odd powers: 9)
26. Which is the only even prime number?
Answer: 2
Explanation:
All other even numbers are divisible by 2.
27. What is the sum of the first 20 natural numbers?
Answer: 210
Explanation:
Sum = n(n+1)/2 = 20 × 21 / 2 = 210
28. Find the LCM of 24 and 36.
Answer: 72
Explanation:
24 = 2³ × 3; 36 = 2² × 3²; LCM = 2³ × 3² = 72
29. If 40% of a number is 80, what is the number?
Answer: 200
Explanation:
0.4x = 80 → x = 200
30. What is the HCF of 18, 24, and 30?
Answer: 6
Explanation:
18 = 2 × 3²; 24 = 2³ × 3; 30 = 2 × 3 × 5; HCF = 2 × 3 = 6
31. What is the unit digit of 8^23?
Answer: 8
Explanation:
Unit digit of 8: 8, 4, 2, 6 (repeats every 4); 23 mod 4 = 3 → 2
32. Find the sum of all odd numbers between 1 and 100.
Answer: 2500
Explanation:
Sum = n², n = 50 (odd numbers); 50² = 2500
33. What is the smallest composite number?
Answer: 4
Explanation:
4 = 2 × 2
34. If the sum of the digits of a number is divisible by 3, what can you say about the number?
Answer: It is divisible by 3
Explanation:
Divisibility rule for 3.
35. What is the HCF of 100 and 140?
Answer: 20
Explanation:
100 = 2² × 5²; 140 = 2² × 5 × 7; HCF = 2² × 5 = 20
36. Find the LCM of 9 and 12.
Answer: 36
Explanation:
9 = 3²; 12 = 2² × 3; LCM = 2² × 3² = 36
37. What is the product of the first five natural numbers?
Answer: 120
Explanation:
1 × 2 × 3 × 4 × 5 = 120
38. Find the sum of all factors of 28.
Answer: 56
Explanation:
Factors: 1, 2, 4, 7, 14, 28; sum = 56
39. What is the unit digit of 3^73?
Answer: 7
Explanation:
Unit digit of 3: 3, 9, 7, 1 (repeats every 4); 73 mod 4 = 1 → 3
40. Which number is neither prime nor composite?
Answer: 1
Explanation:
1 has only one factor.
41. The sum of three consecutive even numbers is 48. What are the numbers?
Answer: 14, 16, 18
Explanation:
Let numbers be x, x+2, x+4; x + x+2 + x+4 = 48 → 3x = 42 → x = 14
42. What is the LCM of 5, 10, and 15?
Answer: 30
Explanation:
LCM of 5, 10, 15 is 30
43. Find the HCF of 27 and 63.
Answer: 9
Explanation:
27 = 3³; 63 = 3² × 7; HCF = 3² = 9
44. If a number is divisible by 2 and 3, is it divisible by 6?
Answer: Yes
Explanation:
LCM of 2 and 3 is 6
45. What is the sum of the first 15 odd numbers?
Answer: 225
Explanation:
Sum = n² = 15² = 225
46. Which is the greatest two-digit prime number?
Answer: 97
Explanation:
Next prime after 97 is 101 (three-digit)
47. Find the LCM of 7 and 9.
Answer: 63
Explanation:
7 × 9 = 63 (since they are co-prime)
48. What is the HCF of 45 and 75?
Answer: 15
Explanation:
45 = 3² × 5; 75 = 3 × 5²; HCF = 3 × 5 = 15
49. What is the unit digit of 2^99?
Answer: 8
Explanation:
Unit digit of 2: 2, 4, 8, 6 (repeats every 4); 99 mod 4 = 3 → 8
50. What is the sum of all prime numbers between 1 and 10?
Answer: 17
Explanation:
2 + 3 + 5 + 7 = 17
Key Takeaways So Far
- Consistency and variety in practice are key to mastery.
- Reviewing solutions is as important as solving problems.
- Tailoring practice to your target exam boosts your chances of success.
Practice and Preparation Resources for Number System
Consistent and focused practice is the best way to master number system questions and excel in quantitative aptitude sections of competitive exams. Here are some effective strategies and resources to guide your preparation:
- Practice with Variety: Work through a wide range of questions, including MCQs, word problems, and conceptual questions. This will help you recognize patterns and develop a deeper understanding of all key topics.
- Use Quality Study Materials: Access questions with answers from reputable books, online platforms, and coaching materials. For those who prefer offline study, download a PDF for easy reference.
- Timed Aptitude Tests: Regularly attempt practice tests to improve your speed and accuracy under exam conditions.
- Placement and Exam Focus: If you’re preparing for specific exams, such as TCS, GATE, or campus placements, seek out resources tailored to those exams. These will reflect the latest exam patterns and difficulty levels.
- Work Through Solutions: Always review detailed solutions. Understanding the logic and shortcuts used in detailed solutions will help you learn tricks and avoid common mistakes.
- Master MCQs and Competitive Sets: Practice multiple-choice and competitive questions to sharpen your problem-solving skills for real exam scenarios.
- Formula Revision: Keep your formulas handy and revise them regularly. This will make it easier to recall and apply them quickly during exams.
- Adaptive and Smart Practice: Use platforms that offer adaptive practice, which tailors question difficulty to your current skill level, ensuring steady progress.
By following these strategies and utilizing quality resources, you’ll build a solid foundation in number system topics and significantly improve your performance. Remember, regular and targeted practice is the key to mastering number system questions and achieving your exam goals.
Conclusion
Mastering number system concepts is a fundamental skill for clearing competitive exams and placement interviews. By practicing a wide range of questions and applying effective tricks, you’ll boost both your confidence and your scores. Start today with comprehensive resources and regular practice—success is within your reach!
Why It Matters
Aptitude on number system is a fundamental skill for clearing competitive exams and placement interviews. Mastery here not only boosts your quantitative scores but also strengthens your overall problem-solving ability.
Practical Advice for Learners
- Practice a wide range of number system questions regularly.
- Memorize and frequently revise essential formulas.
- Focus on understanding concepts, not just rote learning.
- Use adaptive and timed tests to simulate exam environments.
- Analyze solutions for shortcuts and avoid common mistakes.
- Seek out exam-specific question sets for focused preparation.
