⭐ Sabse Important Points · एक नज़र में
- HCF: Greatest common divisor; found by prime factorization or Euclidean algorithm.
- Same remainder: If N divided by a, b, c leaves remainder r, then N - r is a multiple of LCM(a,b,c).
- LCM: Smallest common multiple; found by prime factorization or formula LCM = (a×b)/HCF.
- Different remainders: If N divided by a, b, c leaves remainders r1, r2, r3, then N + (a - r1) = N + (b - r2) = N + (c - r3) is a multiple of LCM.
- Co-prime numbers: HCF = 1; LCM = product of numbers.
- Maximum equal groups: HCF gives the largest size of each group when dividing items into equal groups.
Introduction · परिचय
LCM (Least Common Multiple) and HCF (Highest Common Factor) are fundamental concepts in number theory, frequently tested in SSC MTS. They help solve problems related to multiples, divisors, fractions, and remainders efficiently.
HCF (Highest Common Factor) and LCM (Least Common Multiple)
HCF of two or more numbers is the greatest number that divides each of them exactly. LCM is the smallest number that is a multiple of each of them. For two numbers, HCF × LCM = Product of the numbers. HCF is always a factor of LCM.
📐 Formula / Rule Box · सूत्र याद रखो
Relation between HCF and LCM
HCF(a,b) × LCM(a,b) = a × b
HCF of fractions
HCF(numerators) / LCM(denominators)
LCM of fractions
LCM(numerators) / HCF(denominators)
| HCF | LCM |
|---|---|
| Greatest common divisor | Least common multiple |
| Divides each number | Multiple of each number |
| Always ≤ smallest number | Always ≥ largest number |
| Used in dividing things into smaller parts | Used in finding common meeting times or quantities |
⭐ Key Points · याद रखने वाली बातें
- HCF: Greatest common divisor; found by prime factorization or Euclidean algorithm.
- LCM: Smallest common multiple; found by prime factorization or formula LCM = (a×b)/HCF.
- Co-prime numbers: HCF = 1; LCM = product of numbers.
- HCF of fractions: HCF of numerators / LCM of denominators.
- LCM of fractions: LCM of numerators / HCF of denominators.
✍️ 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 = product of lowest powers = 2×3=6. LCM = product of highest powers = 2²×3²=36. Check: 6×36=216=12×18.
Applications: Remainders and Word Problems
HCF and LCM are used to solve problems involving remainders. If a number leaves the same remainder when divided by several numbers, subtract the remainder from the number to get a multiple of LCM. For HCF, problems involve dividing quantities into maximum equal groups.
⭐ Key Points · याद रखने वाली बातें
- Same remainder: If N divided by a, b, c leaves remainder r, then N - r is a multiple of LCM(a,b,c).
- Different remainders: If N divided by a, b, c leaves remainders r1, r2, r3, then N + (a - r1) = N + (b - r2) = N + (c - r3) is a multiple of LCM.
- Maximum equal groups: HCF gives the largest size of each group when dividing items into equal groups.
- Bells ringing together: LCM of intervals gives the time when they ring together again.
✍️ Solved Example 1 · हल किया हुआ उदाहरण
❓ Question: Find the least number which when divided by 6, 8, and 12 leaves remainder 5 in each case.
✅ Solution: 29
💡 Explanation: LCM(6,8,12) = 24. Required number = 24 + 5 = 29. Check: 29 ÷ 6 = 4 remainder 5, etc.
✍️ Solved Example 2 · हल किया हुआ उदाहरण
❓ Question: Three bells ring at intervals of 4, 6, and 8 minutes. If they ring together at 8:00 AM, when will they next ring together?
✅ Solution: 8:24 AM
💡 Explanation: LCM(4,6,8) = 24 minutes. Next together ring = 8:00 + 24 min = 8:24 AM.
Practice Questions · खुद हल करके देखो
पहले खुद try karo, phir “Answer dekho” dabakar check karo. 👇
Q1Find the HCF of 36 and 84.
👀 Answer dekho
✅ Answer: 12
💡 Explanation: Prime factors: 36 = 2²×3², 84 = 2²×3×7. HCF = 2²×3 = 12.
Q2Find the LCM of 15, 20, and 30.
👀 Answer dekho
✅ Answer: 60
💡 Explanation: Prime factors: 15=3×5, 20=2²×5, 30=2×3×5. LCM = 2²×3×5 = 60.
Q3What is the greatest number that divides 43, 91, and 183 leaving the same remainder in each case?
👀 Answer dekho
✅ Answer: 4
💡 Explanation: Differences: 91-43=48, 183-91=92, 183-43=140. HCF(48,92,140) = 4. So greatest number is 4.
⚠️ Common Mistakes · ये गलतियाँ बिल्कुल मत करना
- Confusing HCF with LCM: HCF is the greatest common divisor, LCM is the least common multiple. Remember HCF ≤ numbers ≤ LCM.
- Forgetting to subtract the remainder when finding the number that leaves the same remainder: N - r is a multiple of LCM.
- Incorrectly applying the formula HCF × LCM = product for more than two numbers: it holds only for two numbers.
⚡ Quick Revision · Exam से पहले 2 मिनट में दोहराओ
- HCF: Greatest common divisor; LCM: Least common multiple.
- For two numbers: HCF × LCM = product of numbers.
- HCF of fractions = HCF of numerators / LCM of denominators.
- LCM of fractions = LCM of numerators / HCF of denominators.
- Same remainder problems: required number = LCM + remainder (if least number) or multiple of LCM + remainder.
- Bells ringing together: time interval = LCM of intervals.