⭐ 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.
- Composite Number: A number greater than 1 with more than two factors. Examples: 4, 6, 8, 9, 10.
- LCM: Product of the highest power of each prime factor involved.
- Divisibility by 3: Sum of digits is divisible by 3.
- Relation: HCF(a, b) × LCM(a, b) = a × b (for two numbers).
Introduction · परिचय
The Number System is a foundational topic for the Delhi Police Constable Numerical Ability section. It covers types of numbers, divisibility rules, factors and multiples, HCF and LCM, and basic number properties. Mastering this topic is essential for quick and accurate problem-solving in arithmetic and simplification questions.
Types of Numbers and Divisibility Rules
Numbers are classified into natural numbers, whole numbers, integers, rational numbers, irrational numbers, prime numbers, composite numbers, even and odd numbers. Divisibility rules help determine if a number is divisible by another without performing full division. For example, a number is divisible by 3 if the sum of its digits is divisible by 3; by 4 if the last two digits form a number divisible by 4; by 9 if the sum of digits is divisible by 9; by 11 if the difference between the sum of digits at odd places and even places is 0 or a multiple of 11.
📐 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.
- Composite Number: A number greater than 1 with more than two factors. Examples: 4, 6, 8, 9, 10.
- Divisibility by 3: Sum of digits is divisible by 3.
- Divisibility by 4: Last two digits form a number divisible by 4.
- Divisibility by 9: Sum of digits is divisible by 9.
- Divisibility by 11: Difference between sum of digits at odd and even places is 0 or multiple of 11.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Check if 483 is divisible by 3.
✅ Solution: Yes, 483 is divisible by 3.
💡 Explanation: Sum of digits = 4 + 8 + 3 = 15, which is divisible by 3. Hence 483 is divisible by 3.
✍️ Solved Example 2 · हल किया हुआ उदाहरण
❓ Question: Check if 1234 is divisible by 4.
✅ Solution: No, 1234 is not divisible by 4.
💡 Explanation: Last two digits form 34, which is not divisible by 4. Hence 1234 is not divisible by 4.
HCF and LCM
HCF (Highest Common Factor) is the largest 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. HCF can be found using prime factorization or the division method. LCM can be found using prime factorization or the formula: LCM(a, b) = (a × b) / HCF(a, b). These concepts are used in problems involving grouping, tiling, and finding common multiples.
📐 Formula / Rule Box · सूत्र याद रखो
HCF-LCM Relation
HCF(a, b) × LCM(a, b) = a × b
LCM by HCF
LCM(a, b) = (a × b) / HCF(a, b)
| HCF | LCM |
|---|---|
| Largest common divisor | Smallest common multiple |
| Used for grouping and distribution | Used for finding common multiples |
| HCF ≤ smallest number | LCM ≥ largest number |
⭐ Key Points · याद रखने वाली बातें
- HCF: Product of the smallest power of each common prime factor.
- LCM: Product of the highest power of each prime factor involved.
- Relation: HCF(a, b) × LCM(a, b) = a × b (for two numbers).
- Co-prime Numbers: Two numbers with HCF = 1. Their LCM is their product.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Find HCF and LCM of 12 and 18.
✅ Solution: HCF = 6, LCM = 36.
💡 Explanation: Prime factors: 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: Find the HCF of 24 and 36 using the division method.
✅ Solution: HCF = 12.
💡 Explanation: Divide 36 by 24: remainder 12. Divide 24 by 12: remainder 0. HCF is the last non-zero remainder, which is 12.
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 factors: 1 and 2. 1 is not prime as it has only one factor.
Q2Find the HCF of 18 and 24.
👀 Answer dekho
✅ Answer: 6
💡 Explanation: Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Common factors: 1, 2, 3, 6. Highest common factor = 6.
Q3Is 572 divisible by 11?
👀 Answer dekho
✅ Answer: Yes
💡 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
- Confusing HCF with LCM: Remember HCF is the largest common divisor, LCM is the smallest common multiple.
- Forgetting that 1 is neither prime nor composite: 1 has only one factor, so it is not prime.
- Misapplying divisibility rule for 4: Check only the last two digits, not the whole number.
- In HCF-LCM relation, using it for more than two numbers: The formula HCF(a,b) × LCM(a,b) = a × b works only for two numbers.
⚡ 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: Largest number that divides two or more numbers exactly.
- LCM: Smallest number that is a multiple of two or more numbers.
- HCF-LCM Relation: For two numbers a and b, HCF(a,b) × LCM(a,b) = a × b.