HCF and LCM: Notes और MCQ Practice

आसान भाषा में:

HCF (Highest Common Factor) वो सबसे बड़ा number है जो दिए गए numbers को completely divide कर दे। LCM (Least Common Multiple) वो सबसे छोटा number है जो दिए गए numbers से completely divide हो जाए। SSC CGL Exam में इससे direct formulas और real-life application-based questions ज़रूर पूछे जाते हैं।

Important Formulas:

• Product Rule: Product of two numbers = HCF × LCM
• HCF of Fractions = HCF of Numerators ÷ LCM of Denominators
• LCM of Fractions = LCM of Numerators ÷ HCF of Denominators
• Ratio Rule: अगर two numbers का ratio a:b है और HCF 'H' है, तो numbers Ha और Hb होंगे, और LCM = H × a × b होगा।

2 Quick Tricks:

1. Difference Method for HCF: HCF हमेशा दिए गए numbers के minimum difference या उस difference का कोई factor होता है। Options को difference से divide करके तुरंत check करें।
2. Option Elimination for LCM: LCM के questions में options को divisor से divide करके remainder check करें, पूरा calculation करने की कोई ज़रूरत नहीं है।

2 Common Mistakes to Avoid:

1. Fraction Simplification: Fractions का HCF/LCM निकालने से पहले उन्हें simplest form (lowest term) में reduce करना मत भूलना, वरना answer गलत आएगा।
2. HCF vs LCM Remainder Rule: HCF के remainder वाले questions में पहले numbers में से remainder minus (-) करते हैं फिर HCF निकालते हैं। LCM में पहले LCM निकालते हैं फिर remainder plus (+) करते हैं। Don't mix them up!

HCF and LCM: Complete Study Notes

This chapter is one of the most scoring areas in the Quantitative Aptitude section of the SSC CGL exam. Questions are asked directly or conceptually integrated with other topics like Time and Work, Pipe and Cisterns, and Number Systems.

1. Concept (Simple Explanation)

A. What is a Factor and a Multiple?

• Factor: A number that divides another number completely without leaving any remainder.
• Example: The factors of 12 are 1, 2, 3, 4, 6, and 12.
• Multiple: A number obtained by multiplying a given number by an integer.
• Example: The multiples of 12 are 12, 24, 36, 48, ...

B. Highest Common Factor (HCF)

HCF (also known as Greatest Common Divisor - GCD) is the largest number that divides two or more given numbers completely.

• Example: Let's find the HCF of 12 and 18.
• Factors of 12: 1, 2, 3, 6, 12
• Factors of 18: 1, 2, 3, 6, 9, 18
• Common factors: 1, 2, 3, 6
• The highest common factor is 6.

C. Least Common Multiple (LCM)

LCM is the smallest positive number that is a perfect multiple of two or more given numbers.

• Example: Let's find the LCM of 12 and 18.
• Multiples of 12: 12, 24, 36, 48, 60, 72, ...
• Multiples of 18: 18, 36, 54, 72, 90, ...
• Common multiples: 36, 72, ...
• The least common multiple is 36.

2. Important Formulas / Rules / Facts

• Product Rule (Only for two numbers):

First Number × Second Number = HCF × LCM

• HCF of Fractions:

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

• LCM of Fractions:

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

• Ratio Rule:

If two numbers are in the ratio a : b and their HCF is H, then:

• The numbers are (H × a) and (H × b).
• Their LCM = H × a × b.
• Co-prime Numbers:

Two numbers are co-prime if their HCF is 1.

• Example: (8, 15). The LCM of two co-prime numbers is always equal to their product.
• HCF of Powers:

HCF of (a^n − 1) and (a^m − 1) = a^(HCF of m, n) − 1

• Remainder Rules for HCF:
• To find the greatest number that divides x, y, and z leaving the same remainder 'r' in each case:

Required Number = HCF of |x − y|, |y − z|, and |z − x|

• To find the greatest number that divides x, y, and z leaving remainders a, b, and c respectively:

Required Number = HCF of (x − a), (y − b), and (z − c)

• Remainder Rules for LCM:
• To find the least number which when divided by x, y, and z leaves a remainder 'r' in each case:

Required Number = (LCM of x, y, z) + r

• To find the least number which when divided by x, y, and z leaves remainders a, b, and c respectively, where the difference (x − a) = (y − b) = (z − c) = k:

Required Number = (LCM of x, y, z) − k

3. Solved Examples (Exam Style)

Example 1: Basic Product Formula

Question: The HCF of two numbers is 11 and their LCM is 693. If one of the numbers is 77, find the other number.

Step-by-step Solution:

1. Let the second number be 'x'.
2. Use the formula: First Number × Second Number = HCF × LCM
3. Substitute the given values:

77 × x = 11 × 693

1. Solve for x:

x = (11 × 693) ÷ 77

x = 693 ÷ 7

x = 99

Answer: The other number is 99.

Example 2: HCF with Different Remainders

Question: Find the greatest number which divides 29, 60, and 103 leaving remainders 5, 12, and 7 respectively.

Step-by-step Solution:

1. Since we need to find the "greatest number which divides", we must calculate the HCF.
2. Subtract the respective remainders from the numbers first:

• 29 − 5 = 24
• 60 − 12 = 48
• 103 − 7 = 96

1. Now, find the HCF of 24, 48, and 96.

• Factors of 24 = 24 × 1
• Factors of 48 = 24 × 2
• Factors of 96 = 24 × 4

1. The highest common factor among 24, 48, and 96 is 24.

Answer: The required greatest number is 24.

Example 3: LCM Application (Bells/Traffic Lights)

Question: Three bells toll together at intervals of 9, 12, and 15 minutes respectively. If they toll together now, after how many hours will they toll together next?

Step-by-step Solution:

1. For interval-based repeating events, we always find the LCM of the given intervals.
2. Find the LCM of 9, 12, and 15:

• Prime factorization of 9 = 3²
• Prime factorization of 12 = 2² × 3
• Prime factorization of 15 = 3 × 5

1. LCM = 2² × 3² × 5 = 4 × 9 × 5 = 180 minutes.
2. Convert minutes into hours:

180 minutes ÷ 60 = 3 hours.

Answer: The bells will toll together next after 3 hours.

Example 4: LCM with Constant Difference Remainder

Question: Find the least number which when divided by 20, 25, and 35 leaves remainders 14, 19, and 29 respectively.

Step-by-step Solution:

1. Observe the difference between the divisors and their respective remainders:

• 20 − 14 = 6
• 25 − 19 = 6
• 35 − 29 = 6

Since the difference 'k' is constant (k = 6), we use the formula: Required Number = (LCM of divisors) − k.

1. Find the LCM of 20, 25, and 35:

• 20 = 2² × 5
• 25 = 5²
• 35 = 5 × 7
• LCM = 2² × 5² × 7 = 4 × 25 × 7 = 700

1. Subtract the constant difference (k = 6):

Required Number = 700 − 6 = 694.

Answer: The required least number is 694.

4. Shortcuts and Tricks

Trick 1: Finding HCF using the Difference Method

Instead of long division or prime factorization, you can find the HCF of numbers by calculating the difference between them.

• Rule: The HCF of two or more numbers is either their difference or a factor of their difference.
• Example: Find HCF of 306 and 391.
• Difference = 391 − 306 = 85.
• Factors of 85 are 1, 5, 17, 85.
• Check if 85 divides 306 (No, because 306 does not end in 5 or 0).
• Check the next largest factor, 17. Does 17 divide 306? Yes (17 × 18 = 306). Does 17 divide 391? Yes (17 × 23 = 391).
• Therefore, HCF is 17.

Trick 2: Option Elimination Method (Highly Recommended for SSC CGL)

• For LCM Questions: The correct option must be completely divisible by all the numbers given in the question. Test divisibility rules (especially for 3, 9, 5, and 11) on the options to eliminate wrong answers in seconds.
• For HCF Questions: The correct option must completely divide all the numbers given in the question. Start checking from the largest option down to the smallest.

5. Common Mistakes to Avoid

• Mistake 1: Applying the Product Rule to Three Numbers
• Incorrect: Number 1 × Number 2 × Number 3 = HCF × LCM.
• Correct: The formula Product = HCF × LCM is strictly valid only for two numbers.
• Mistake 2: Confusing the Remainder Steps between HCF and LCM
• In HCF problems: Subtract the remainders before calculating the HCF.
• In LCM problems: Calculate the LCM first, and then add or subtract the remainder value at the end.
• Mistake 3: Fraction Formula Inversion
• Students often calculate the LCM of the denominators when asked for the LCM of fractions. Always remember: whatever you need to find (HCF or LCM), perform that operation on the numerators and the opposite operation on the denominators.