LCM: Notes और MCQ Practice

आसान भाषा में

LCM यानी Least Common Multiple वो smallest number होता है जो दिए गए सभी numbers से completely divide (विभाजित) हो जाए। सरल शब्दों में, ऐसा सबसे छोटा common multiple जो सबके table में आए। RRB NTPC में Traffic lights, Circular track running, और Time & Work के सवाल direct LCM पर ही solve होते हैं!

Important Formulas & Rules

• Two Numbers Relation: First Number × Second Number = LCM × HCF
• LCM of Fractions: (LCM of Numerators) ÷ (HCF of Denominators)
• Co-prime Numbers: अगर numbers Co-prime हैं (जिनका HCF = 1 हो), तो उनका LCM हमेशा उनका Product ही होगा।
• Remainder Concept: वो smallest number जिसे x, y, z से divide करने पर 'r' remainder बचे = (LCM of x, y, z) + r

2 Quick Exam Tricks

1. Largest Number Multiple Trick: Pen चलाने की ज़रूरत नहीं! सबसे बड़े number का table मन में पढ़ें।

• Example: 6, 8, 12 का LCM चाहिए? सबसे बड़ा number 12 है। 12 × 1 = 12 (8 से divide नहीं होता), 12 × 2 = 24 (6 और 8 दोनों से divide होता है) → LCM = 24.

1. Factor Elimination: अगर list में कोई छोटा number किसी बड़े number को divide कर रहा है, तो छोटे number को ignore कर दो।

• Example: 4, 8, 16, 32 में 4, 8, 16 को सीधे काट दो क्योंकि 32 में वो already शामिल हैं → LCM = 32.

2 Common Mistakes

1. Fractions Rule Confusion: Fraction का LCM निकालते समय students अक्सर denominator का भी LCM ले लेते हैं। हमेशा याद रखें: Fraction के LCM में Numerators का LCM और Denominators का HCF होता है।
2. Formula Overuse: Product of numbers = LCM × HCF वाला rule सिर्फ 2 numbers के लिए valid है। इसे 3 या उससे ज़्यादा numbers पर direct apply न करें!

LCM (Lowest Common Multiple) - Complete Study Notes

1. Concept

The Lowest Common Multiple (LCM) of two or more numbers is the smallest non-zero common number that is a multiple of each of the given numbers. In simple words, it is the smallest number that can be divided evenly (leaving no remainder) by all the given numbers.

• What is a multiple? Multiples of a number are the numbers obtained by multiplying it with 1, 2, 3, 4, etc.
• Multiples of 4: 4, 8, 12, 16, 20, 24, 28, ...
• Multiples of 6: 6, 12, 18, 24, 30, ...
• Common Multiples: 12, 24, 36, ...
• Lowest Common Multiple (LCM): The smallest among them is 12.
• Therefore, LCM(4, 6) = 12.

2. Important Formulas, Rules, and Facts

1. Basic Relationship for Two Numbers:

• Product of two numbers = LCM × HCF
• First Number × Second Number = LCM × HCF

(Note: This rule directly applies only to two numbers, not generally to three or more numbers).

1. LCM of Fractions:

• LCM of Fractions = LCM of Numerators ÷ HCF of Denominators

1. Co-prime Numbers:

• If two numbers have no common factor other than 1 (e.g., 8 and 9), their LCM is simply their product: LCM(a, b) = a × b.

1. Factor-Multiple Rule:

• If a smaller number divides a larger number completely, the larger number is their LCM.
• Example: LCM(6, 18) = 18.

1. Standard Word Problem Models:

• Model 1 (Same Remainder 'R'): The least number which when divided by x, y, and z leaves remainder 'R' in each case = [LCM(x, y, z)] + R.
• Model 2 (Different Remainders with Constant Difference 'K'): The least number which when divided by x, y, and z leaves remainders a, b, and c respectively, such that (x − a) = (y − b) = (z − c) = K = [LCM(x, y, z)] − K.
• Model 3 (Simultaneous Events): Problems involving bells ringing together, runners running around a circular track, or traffic lights blinking simultaneously require finding the LCM of their respective individual time intervals.

3. Solved Examples

Example 1: Finding LCM by Division Method

Question: Find the LCM of 12, 18, and 30.

Step-by-step Solution:

1. Write down the numbers in a row and divide by the smallest prime number that divides at least one of them:

• Divide by 2: (12, 18, 30) → (6, 9, 15)
• Divide by 2: (6, 9, 15) → (3, 9, 15)
• Divide by 3: (3, 9, 15) → (1, 3, 5)
• Divide by 3: (1, 3, 5) → (1, 1, 5)
• Divide by 5: (1, 1, 5) → (1, 1, 1)

1. Multiply all divisors:

• LCM = 2 × 2 × 3 × 3 × 5
• LCM = 4 × 9 × 5 = 180.

Answer: 180

Example 2: LCM of Fractions

Question: Find the LCM of 2/3, 8/9, and 16/81.

Step-by-step Solution:

1. Formula: LCM of Fractions = LCM of Numerators ÷ HCF of Denominators
2. Numerators are 2, 8, 16:

• Since 16 is divisible by both 2 and 8, LCM(2, 8, 16) = 16.

1. Denominators are 3, 9, 81:

• The largest common factor dividing 3, 9, and 81 is 3. So, HCF(3, 9, 81) = 3.

1. Apply formula:

• LCM = 16 ÷ 3 = 16/3.

Answer: 16/3

Example 3: Simultaneous Events (Bells Ringing)

Question: Three church bells ring at intervals of 10, 15, and 20 minutes respectively. If all three ring together at 8:00 AM, at what time will they next ring together?

Step-by-step Solution:

1. Find the LCM of the intervals: 10, 15, and 20 minutes.

• Prime factors of 10 = 2 × 5
• Prime factors of 15 = 3 × 5
• Prime factors of 20 = 2² × 5
• LCM = 2² × 3 × 5 = 4 × 3 × 5 = 60 minutes.

1. The bells ring together every 60 minutes (or 1 hour).
2. Add 1 hour to the initial time (8:00 AM):

• 8:00 AM + 1 hour = 9:00 AM.

Answer: 9:00 AM

Example 4: Number Leaving a Specific Remainder

Question: Find the least number which when divided by 12, 16, and 24 leaves a remainder 7 in each case.

Step-by-step Solution:

1. This follows Model 1: Required Number = LCM(12, 16, 24) + Remainder.
2. Find LCM(12, 16, 24):

• Multiples of 24: 24 (not div by 16), 48 (divisible by 12, 16, and 24).
• Hence, LCM = 48.

1. Add the remainder:

• Required number = 48 + 7 = 55.

1. Verification: 55 ÷ 12 gives quotient 4, remainder 7. Correct.

Answer: 55

Example 5: Constant Difference Remainder

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

Step-by-step Solution:

1. Check the difference between divisor and remainder:

• 20 − 14 = 6
• 25 − 19 = 6
• 35 − 29 = 6
• The common difference (K) = 6.

1. Formula: Required Number = LCM(20, 25, 35) − K.
2. Find LCM(20, 25, 35):

• 20 = 2² × 5
• 25 = 5²
• 35 = 5 × 7
• LCM = 2² × 5² × 7 = 4 × 25 × 7 = 100 × 7 = 700.

1. Subtract constant difference (K = 6):

• Required number = 700 − 6 = 694.

Answer: 694

4. Shortcuts and Tricks

1. The 'Drop the Factors' Shortcut:

• If you need the LCM of 4, 8, 12, and 24:
• Since 4 divides 24 completely, ignore 4.
• Since 8 divides 24 completely, ignore 8.
• Since 12 divides 24 completely, ignore 12.
• The LCM is instantly 24.

1. The 'Highest Power' Prime Method:

• Break numbers into prime factors and take the highest power of every prime that appears.
• For 18 (2¹ × 3²) and 24 (2³ × 3¹):
• Highest power of 2 is 2³ (8); highest power of 3 is 3² (9).
• LCM = 8 × 9 = 72.

1. Using Options Elimination in the Exam:

• In multiple-choice questions, instead of full calculation, use divisibility rules (like 3, 9, 11).
• Example: If 9 is one of the divisors, the correct LCM must have a sum of digits divisible by 9.

5. Common Mistakes to Avoid

• Confusing LCM with HCF: Remember that LCM is a multiple—it is always equal to or greater than the largest number in the group. HCF is a factor—it is always equal to or smaller than the smallest number.
• Flipping the Fraction Rule: Students often calculate HCF of Numerators ÷ LCM of Denominators when asked for LCM. For LCM of fractions, LCM always stays in the numerator.
• Adding vs. Subtracting Remainders:
• When remainder is the same for all divisors: ADD the remainder to the LCM.
• When divisor − remainder = constant difference: SUBTRACT that constant difference from the LCM.
• Extending 'Product = LCM × HCF' to Three Numbers: Do not use Number1 × Number2 × Number3 = LCM × HCF. This equation is invalid for three numbers.