⭐ Sabse Important Points · एक नज़र में
- Natural Numbers: Counting numbers starting from 1 (1, 2, 3, ...).
- Factor: A number that divides another number exactly (e.g., factors of 12: 1,2,3,4,6,12).
- Whole Numbers: Natural numbers including 0 (0, 1, 2, ...).
- Multiple: Product of a number with an integer (e.g., multiples of 5: 5,10,15,...).
- Integers: All positive and negative whole numbers including 0.
- HCF: Product of the smallest power of each common prime factor.
Introduction · परिचय
The Number System is a fundamental topic in the Agniveer Army GD Mathematics syllabus. It covers types of numbers, divisibility rules, factors and multiples, HCF and LCM, and basic number properties. A strong grasp of this topic is essential for quick calculations and solving arithmetic problems accurately.
Types of Numbers and Divisibility Rules
Numbers are classified into natural numbers, whole numbers, integers, rational numbers, irrational numbers, prime numbers, and composite numbers. Divisibility rules help determine whether a number is divisible by another without performing full division. These rules are crucial for simplifying fractions, finding factors, and solving problems quickly.
- Check last digit for 2,5,10
- Check sum of digits for 3,9
- Check last two digits for 4
- Check alternating sum for 11
⭐ Key Points · याद रखने वाली बातें
- Natural Numbers: Counting numbers starting from 1 (1, 2, 3, ...).
- Whole Numbers: Natural numbers including 0 (0, 1, 2, ...).
- Integers: All positive and negative whole numbers including 0.
- Prime Number: A number greater than 1 with exactly two factors: 1 and itself (e.g., 2, 3, 5, 7).
- Composite Number: A number greater than 1 with more than two factors (e.g., 4, 6, 8, 9).
- Divisibility by 2: Last digit is even (0, 2, 4, 6, 8).
- Divisibility by 3: Sum of digits is divisible by 3.
- Divisibility by 4: Last two digits form a number divisible by 4.
- Divisibility by 5: Last digit is 0 or 5.
- Divisibility by 9: Sum of digits is divisible by 9.
- Divisibility by 11: Difference between sum of digits at odd places and sum of digits at even places is 0 or a multiple of 11.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Check if 3,528 is divisible by 3, 4, and 9.
✅ Solution: Divisible by 3 and 9, not by 4.
💡 Explanation: Sum of digits = 3+5+2+8=18, which is divisible by 3 and 9. Last two digits = 28, which is not divisible by 4. Hence, 3528 is divisible by 3 and 9 but not by 4.
✍️ Solved Example 2 · हल किया हुआ उदाहरण
❓ Question: Is 1,234 divisible by 11?
✅ Solution: No.
💡 Explanation: Sum of digits at odd places (from right): 4+2=6; sum at even places: 3+1=4; difference = 6-4=2, not a multiple of 11. So 1234 is not divisible by 11.
Factors, Multiples, HCF and LCM
Factors of a number divide it exactly without leaving a remainder. Multiples are obtained by multiplying the number by integers. 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. HCF and LCM are used in simplifying fractions and solving problems on meeting points, tiling, and grouping.
📐 Formula / Rule Box · सूत्र याद रखो
HCF-LCM Relation
HCF(a,b) × LCM(a,b) = a × b
LCM of Fractions
LCM of fractions = LCM of numerators / HCF of denominators
HCF of Fractions
HCF of fractions = HCF of numerators / LCM of denominators
| HCF | LCM |
|---|---|
| Greatest common divisor | Smallest common multiple |
| Used to simplify fractions | Used to find common denominators |
| HCF ≤ smaller number | LCM ≥ larger number |
| Product of common prime factors with least powers | Product of all prime factors with highest powers |
⭐ Key Points · याद रखने वाली बातें
- Factor: A number that divides another number exactly (e.g., factors of 12: 1,2,3,4,6,12).
- Multiple: Product of a number with an integer (e.g., multiples of 5: 5,10,15,...).
- HCF: Product of the smallest power of each common prime factor.
- LCM: Product of the highest power of each prime factor involved.
- Co-prime Numbers: Two numbers with HCF = 1 (e.g., 8 and 15).
- Relation: For two numbers a and b, HCF × LCM = a × b.
✍️ 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: Two numbers are in ratio 2:3 and their HCF is 6. Find the numbers.
✅ Solution: 12 and 18.
💡 Explanation: Let numbers be 2x and 3x. HCF = x = 6. So numbers are 12 and 18.
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 24 and 36.
👀 Answer dekho
✅ Answer: 12
💡 Explanation: Prime factors: 24 = 2³ × 3, 36 = 2² × 3². HCF = 2² × 3 = 12.
Q3Is 4,563 divisible by 9?
👀 Answer dekho
✅ Answer: Yes
💡 Explanation: Sum of digits = 4+5+6+3=18, which is divisible by 9. Hence 4563 is divisible by 9.
Q4The LCM of two numbers is 120 and their HCF is 10. If one number is 30, find the other.
👀 Answer dekho
✅ Answer: 40
💡 Explanation: Using HCF × LCM = product of numbers: 10 × 120 = 30 × x ⇒ 1200 = 30x ⇒ x = 40.
⚠️ Common Mistakes · ये गलतियाँ बिल्कुल मत करना
- Confusing factors with multiples: Factors divide the number exactly, while multiples are obtained by multiplying the number.
- Forgetting that 1 is neither prime nor composite.
- In divisibility by 4, checking only the last digit instead of the last two digits.
- In HCF/LCM word problems, misinterpreting whether to find HCF or LCM; read the problem carefully (e.g., 'greatest number that divides' indicates HCF, 'smallest number divisible by' indicates LCM).
- Assuming all odd numbers are prime; e.g., 9, 15, 21 are odd but composite.
⚡ Quick Revision · Exam से पहले 2 मिनट में दोहराओ
- Prime Numbers: 2 is the only even prime; 1 is neither prime nor composite.
- Divisibility Rules: Memorize rules for 2,3,4,5,9,11 for quick checks.
- HCF and LCM: HCF × LCM = product of two numbers (valid for two numbers only).
- Co-prime Numbers: HCF = 1; e.g., 8 and 15.
- Number of Factors: For N = a^p × b^q, number of factors = (p+1)(q+1).