HCF: Notes और MCQ Practice

आसान भाषा में:

HCF यानी Highest Common Factor वह सबसे बड़ा Number होता है जो दिए गए सभी Numbers को बिना किसी Remainder के completely divide कर दे। इसे GCD (Greatest Common Divisor) भी कहा जाता है। उदाहरण के लिए, 12 और 18 का HCF 6 है, क्योंकि 6 सबसे बड़ा Number है जो दोनों को पूरा-पूरा divide करता है।

Important Formulas & Rules:

• Product Rule: First Number × Second Number = HCF × LCM (ध्यान रहे: यह सिर्फ two numbers के लिए valid है)
• HCF of Fractions: (HCF of Numerators) ÷ (LCM of Denominators)
• Co-prime Numbers का HCF: हमेशा 1 होता है (जैसे 8 और 15 का HCF = 1).
• Divisibility Rule: HCF हमेशा दिए गए numbers के LCM को completely divide करता है।

2 Quick Exam Tricks:

1. Difference Method (जादुई ट्रिक): दिए गए numbers में से किन्हीं दो सबसे नजदीकी numbers का Difference निकालें। HCF या तो वही Difference होगा, या फिर उस Difference का कोई Factor होगा।

• Example: 48 और 60 का difference = 12. 12 दोनों को divide करता है, इसलिए HCF = 12.

1. Option Elimination: HCF के questions में हमेशा Options को Descending Order (बड़े से छोटे) में check करें। जो सबसे बड़ा option सभी numbers को divide कर दे, वही आपका answer है!

2 Common Mistakes (सावधान रहें!):

1. Fractions में गलती: अक्सर students HCF निकालते वक्त Denominator का भी HCF निकाल देते हैं। याद रखें: Numerator का HCF और Denominator का LCM निकालना होता है।
2. Formula का गलत use: Number 1 × Number 2 × Number 3 = HCF × LCM कभी मत करना! यह Formula 3 या उससे ज्यादा numbers पर directly apply नहीं होता।

HCF (Highest Common Factor)

1. Concept

The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of two or more non-zero integers is the largest positive integer that divides each of the numbers without leaving a remainder.

• Factor: A number that divides another number completely without any remainder. For example, factors of 12 are 1, 2, 3, 4, 6, and 12.
• Common Factor: A factor shared by two or more numbers.
• Highest Common Factor: The largest among those common factors.

Simple Example:

• Factors of 12 = 1, 2, 3, 4, 6, 12
• Factors of 18 = 1, 2, 3, 6, 9, 18
• Common Factors = 1, 2, 3, 6
• HCF(12, 18) = 6

2. Important Formulas, Rules, and Facts

• HCF of Fractions:

HCF of fractions = (HCF of Numerators) ÷ (LCM of Denominators)

• Relationship between HCF and LCM (For two numbers only):

First Number × Second Number = HCF × LCM

(Note: This rule does not directly apply to three or more numbers.)

• Co-prime Numbers:

Two numbers are said to be co-prime if their HCF is 1 (e.g., 8 and 15).

• Divisibility Properties:
• The HCF of two or more numbers always divides their LCM completely.
• The HCF of two numbers also divides their sum, difference, and product.
• The HCF of a given set of numbers can never be greater than the smallest number in the set.

• Standard Remainder Cases:
• Case 1: The greatest number that divides numbers x, y, and z leaving remainders a, b, and c respectively = HCF of (x − a), (y − b), and (z − c).
• Case 2: The greatest number that divides numbers x, y, and z leaving the same remainder 'r' in each case = HCF of (x − r), (y − r), and (z − r).
• Case 3: The greatest number that divides numbers x, y, and z leaving the same remainder in each case (when remainder 'r' is not given) = HCF of |x − y|, |y − z|, and |z − x|.

3. Solved Examples (Exam Style)

Example 1: Basic HCF Calculation

Question: Find the HCF of 144, 180, and 192.

Solution:

• Step 1: Write prime factorizations:
• 144 = 2⁴ × 3² = 2 × 2 × 2 × 2 × 3 × 3
• 180 = 2² × 3² × 5 = 2 × 2 × 3 × 3 × 5
• 192 = 2⁶ × 3¹ = 2 × 2 × 2 × 2 × 2 × 2 × 3
• Step 2: Take the lowest power of each common prime factor:
• Common prime factors are 2 and 3.
• Lowest power of 2 is 2².
• Lowest power of 3 is 3¹.
• Step 3: Multiply them together:
• HCF = 2² × 3¹ = 4 × 3 = 12

Answer: 12

Example 2: HCF of Fractions

Question: Find the HCF of 2/3, 8/9, 16/81, and 10/27.

Solution:

• Step 1: Identify numerators and denominators:
• Numerators: 2, 8, 16, 10
• Denominators: 3, 9, 81, 27
• Step 2: Apply the formula:
• HCF of fractions = (HCF of Numerators) ÷ (LCM of Denominators)
• Step 3: Calculate HCF of numerators (2, 8, 16, 10):
• The largest number dividing 2, 8, 16, and 10 is 2.
• Step 4: Calculate LCM of denominators (3, 9, 81, 27):
• Since 81 is a multiple of 3, 9, and 27, LCM = 81.
• Step 5: Combine values:
• HCF = 2/81

Answer: 2/81

Example 3: Different Remainders Case

Question: What is the greatest number that divides 1354, 1866, and 2762 leaving a remainder of 10 in each case?

Solution:

• Step 1: Subtract the remainder from each number:
• 1354 − 10 = 1344
• 1866 − 10 = 1856
• 2762 − 10 = 2752
• Step 2: Find HCF of 1344, 1856, and 2752:
• Use difference between smallest pairs: 1856 − 1344 = 512
• Factors of 512 = 512, 256, 128, 64, 32...
• Check 64:
• 1344 ÷ 64 = 21 (Divisible)
• 1856 ÷ 64 = 29 (Divisible)
• 2752 ÷ 64 = 43 (Divisible)

Answer: 64

Example 4: Same Remainder (Unknown Remainder)

Question: Find the largest number that divides 62, 132, and 237 leaving the same remainder in each case.

Solution:

• Step 1: Find positive differences between the pairs:
• |132 − 62| = 70
• |237 − 132| = 105
• |237 − 62| = 175
• Step 2: Find the HCF of 70, 105, and 175:
• 70 = 35 × 2
• 105 = 35 × 3
• 175 = 35 × 5
• Step 3: The highest common factor is 35.

Answer: 35

4. Shortcuts and Tricks

1. The Difference Method (Fastest for Exams):

• The HCF of two or more numbers is always equal to the difference between the two closest numbers, or it is a factor of that difference.
• Example: Find HCF of 306 and 391.
• Difference = 391 − 306 = 85.
• Factors of 85 are 1, 5, 17, 85.
• 85 does not divide 306 (last digit not 0 or 5). 5 does not divide 306.
• Check 17: 306 ÷ 17 = 18; 391 ÷ 17 = 23.
• Therefore, HCF = 17.

1. Back-solving via Options:

• Since HCF means the Highest Common Factor, start testing options from the largest value to the smallest.
• The first option that divides all the given numbers completely is your answer.

1. Ratio of Numbers:

• If two numbers are in the ratio a : b (where a and b are co-prime) and their HCF is H, then:
• First number = a × H
• Second number = b × H
• LCM = a × b × H

5. Common Mistakes to Avoid

• Confusing HCF and LCM in Word Problems:
• Words like "greatest size", "maximum length", "largest measuring tape", or "divided into equal parts" indicate HCF.
• Words like "minimum time", "ringing together", or "meeting again at the starting point" indicate LCM.
• Misapplying Product Rule:
• Formula Number 1 × Number 2 = HCF × LCM is strictly true for two numbers. Do not use A × B × C = HCF × LCM for three numbers.
• Inverting the Fraction Formula:
• Common error: Calculating (LCM of Numerators) ÷ (HCF of Denominators) when asked for HCF.
• Remember: For HCF of fractions, the numerator gets HCF.
• Forgetting to Subtract Remainders First:
• In remainder problems, always subtract the remainder from the numbers before finding the HCF. Calculating HCF first and subtracting remainder later gives an incorrect result.