⭐ Sabse Important Points · एक नज़र में
- Prime Number: A number greater than 1 with exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, ...
- HCF: Product of the smallest power of each common prime factor in the numbers.
- Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
- LCM: Product of the highest power of each prime factor involved in the numbers.
- Divisibility by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
- Relation: For two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b.
Introduction · परिचय
The Number System is a foundational topic for RRB NTPC Mathematics. It covers types of numbers, divisibility rules, factors and multiples, HCF and LCM, and basic number properties. Mastery of these concepts is essential for quick and accurate problem-solving in the exam.
Types of Numbers and Divisibility Rules
Numbers are classified into natural numbers, whole numbers, integers, rational numbers, irrational numbers, prime numbers, composite numbers, etc. Divisibility rules help determine whether a number is divisible by another without performing full division. These rules are crucial for simplification and number-based problems.
📐 Formula / Rule Box · सूत्र याद रखो
Divisibility by 3
Sum of digits divisible by 3
Divisibility by 4
Last two digits divisible by 4
Divisibility by 9
Sum of digits divisible by 9
Divisibility by 11
(Sum of digits at odd places) - (Sum of digits at even places) = 0 or multiple of 11
⭐ Key Points · याद रखने वाली बातें
- Prime Number: A number greater than 1 with exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, ...
- Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
- Divisibility by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
- Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
- Divisibility by 11: A number is divisible by 11 if the difference between the sum of digits at odd places and sum of digits at even places is either 0 or a multiple of 11.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Check whether 483 is divisible by 3.
✅ Solution: Yes, 483 is divisible by 3.
💡 Explanation: Sum of digits = 4 + 8 + 3 = 15. Since 15 is divisible by 3, 483 is divisible by 3.
✍️ Solved Example 2 · हल किया हुआ उदाहरण
❓ Question: Check whether 1234 is divisible by 4.
✅ Solution: No, 1234 is not divisible by 4.
💡 Explanation: Last two digits form 34. 34 is not divisible by 4 (since 4 × 8 = 32, 4 × 9 = 36). Hence 1234 is not divisible by 4.
HCF and LCM
HCF (Highest Common Factor) is the greatest number that divides two or more numbers exactly. LCM (Lowest Common Multiple) is the smallest number that is a multiple of two or more numbers. These concepts are used in problems involving grouping, distribution, and finding common multiples.
📐 Formula / Rule Box · सूत्र याद रखो
HCF-LCM Relation
HCF(a, b) × LCM(a, b) = a × b
LCM of Co-prime Numbers
LCM(a, b) = a × b if HCF(a, b) = 1
| HCF | LCM |
|---|---|
| Greatest common divisor | Smallest common multiple |
| Used for dividing into smaller groups | Used for finding common meeting times |
| HCF ≤ min(a, b) | LCM ≥ max(a, b) |
⭐ Key Points · याद रखने वाली बातें
- HCF: Product of the smallest power of each common prime factor in the numbers.
- LCM: Product of the highest power of each prime factor involved in the numbers.
- Relation: For two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b.
- Co-prime numbers: Two numbers are co-prime if their HCF is 1. For co-prime numbers, LCM is their product.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Find the HCF and LCM of 12 and 18.
✅ Solution: HCF = 6, LCM = 36.
💡 Explanation: Prime factorization: 12 = 2² × 3, 18 = 2 × 3². HCF = 2¹ × 3¹ = 6. LCM = 2² × 3² = 36. Check: 6 × 36 = 216 = 12 × 18.
✍️ Solved Example 2 · हल किया हुआ उदाहरण
❓ Question: The HCF of two numbers is 4 and their LCM is 48. If one number is 12, find the other.
✅ Solution: The other number is 16.
💡 Explanation: Using HCF × LCM = product of numbers: 4 × 48 = 12 × x ⇒ 192 = 12x ⇒ x = 16.
Practice Questions · खुद हल करके देखो
पहले खुद try karo, phir “Answer dekho” dabakar check karo. 👇
Q1What is the smallest prime number?
👀 Answer dekho
✅ Answer: 2
💡 Explanation: 2 is the smallest prime number because it has exactly two distinct factors: 1 and 2. 1 is not prime as it has only one factor.
Q2Find the HCF of 24 and 36.
👀 Answer dekho
✅ Answer: 12
💡 Explanation: Prime factorization: 24 = 2³ × 3, 36 = 2² × 3². HCF = product of smallest powers of common factors = 2² × 3 = 12.
Q3Check whether 572 is divisible by 11.
👀 Answer dekho
✅ Answer: Yes, 572 is divisible by 11.
💡 Explanation: Sum of digits at odd places (from right): 2 + 5 = 7. Sum of digits at even places: 7. Difference = 7 - 7 = 0. Since difference is 0, 572 is divisible by 11.
⚠️ Common Mistakes · ये गलतियाँ बिल्कुल मत करना
- Mistake: Considering 1 as a prime number. Correction: 1 is neither prime nor composite; it has only one factor.
- Mistake: Forgetting to check divisibility by 8 using the last three digits. Correction: A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
- Mistake: Using HCF × LCM = product for more than two numbers. Correction: This relation holds only for two numbers. For more than two, use prime factorization.
⚡ Quick Revision · Exam से पहले 2 मिनट में दोहराओ
- Prime Number: A number greater than 1 with exactly two factors: 1 and itself.
- Divisibility by 3: Sum of digits divisible by 3.
- HCF-LCM Relation: For two numbers a and b, HCF(a, b) × LCM(a, b) = a × b.
- Co-prime Numbers: Two numbers with HCF 1; their LCM is their product.