**CAT Number System Mastery: LCM, HCF & Cyclicity**

CAT Number System Mastery: LCM, HCF & Cyclicity

(A Premium Study Guide for 99+ Percentile Aspirants)

What This Is

The Number System (LCM, HCF, Cyclicity) is a high-frequency, high-scoring topic in CAT QA. It appears in 3–5 questions per paper, often in TITA (Type In The Answer) format, making it a must-master area. These questions test logical reasoning, pattern recognition, and quick mental math—skills that separate 95th-percentile scorers from 99+.

Real-CAT Example:
What is the last digit of ( 7^{2023} + 3^{2023} )? (Answer: 0. Uses cyclicity of 7 and 3, and LCM properties.)

Mastering this topic ensures: ✅ Accuracy (no silly calculation errors) ✅ Speed (solve in <2 minutes per question) ✅ Confidence (even in seemingly complex problems)

Key Concepts & Techniques

1. LCM & HCF Basics

• LCM (Least Common Multiple): Smallest number divisible by all given numbers.
• When to use: Problems involving repetition, cycles, or synchronization (e.g., bells ringing together, lights flashing).
• HCF (Highest Common Factor): Largest number that divides all given numbers.
• When to use: Problems involving division, splitting, or grouping (e.g., cutting ropes into equal pieces).

Formula:
[ \text{LCM}(a, b) \times \text{HCF}(a, b) = a \times b ] (Only for two numbers! For >2 numbers, use prime factorization.)

2. Prime Factorization (PF) Method

• Break numbers into prime factors (e.g., ( 12 = 2^2 \times 3 )).
• LCM: Take the highest power of each prime.
• HCF: Take the lowest power of each prime.
• When to use: All LCM/HCF problems, especially with 3+ numbers.

Example:
Find LCM & HCF of 12, 18, 24.
- ( 12 = 2^2 \times 3 ) - ( 18 = 2 \times 3^2 ) - ( 24 = 2^3 \times 3 ) - LCM: ( 2^3 \times 3^2 = 72 ) - HCF: ( 2 \times 3 = 6 )

3. Cyclicity of Numbers (Last Digit Patterns)

• Last digit of powers repeats in cycles (e.g., 2: 2,4,8,6 → cycle of 4).
• Key Cycles:
• 2, 3, 7, 8: Cycle of 4
• 4, 9: Cycle of 2
• 5, 6: Always 5, 6 (cycle of 1)
• 0,1: Always 0,1 (cycle of 1)
• When to use: Last digit problems (e.g., ( 7^{2023} ), ( 3^{100} )).

Trick:
To find last digit of ( a^b ): 1. Find ( b \mod \text{cycle length} ).
2. If remainder = 0, take the last digit of the cycle.
3. Else, take the remainder-th digit of the cycle.

Example:
Last digit of ( 7^{2023} ): - Cycle of 7: 7,9,3,1 (length 4).
- ( 2023 \mod 4 = 3 ).
- 3rd digit in cycle: 3.

4. LCM/HCF of Fractions

• LCM of fractions: ( \frac{\text{LCM of numerators}}{\text{HCF of denominators}} )
• HCF of fractions: ( \frac{\text{HCF of numerators}}{\text{LCM of denominators}} )
• When to use: Problems involving fractions or ratios (e.g., "Find the smallest fraction divisible by ( \frac{2}{3} ) and ( \frac{3}{4} )").

Example:
LCM of ( \frac{2}{3} ) and ( \frac{3}{4} ): - LCM(2,3) = 6 - HCF(3,4) = 1 - LCM = ( \frac{6}{1} = 6 )

5. LCM/HCF with Remainders

• If a number leaves remainder ( r ) when divided by ( n ), it is of the form ( kn + r ).
• When to use: Problems like "Find the smallest number that leaves remainder 2 when divided by 3, 4, and 5."

Approach:
1. Find LCM of divisors (3,4,5 → LCM = 60).
2. Add remainder (2) → 60 + 2 = 62.

6. Co-prime Numbers & LCM/HCF

• Co-prime numbers: HCF = 1.
• If two numbers are co-prime, LCM = product of the numbers.
• When to use: Problems involving relative primes (e.g., "Find LCM of 8 and 15").

Example:
LCM(8,15) = 8 × 15 = 120 (since HCF(8,15) = 1).

7. LCM/HCF of Large Numbers (Shortcut)

• For large numbers, use the formula:
  [ \text{LCM}(a, b) = \frac{a \times b}{\text{HCF}(a, b)} ]
• When to use: When prime factorization is tedious (e.g., LCM of 12345 and 67890).

Step-by-Step Strategy

Step 1: Identify the Problem Type

• Last digit? → Cyclicity
• LCM/HCF of numbers? → Prime factorization or formula
• Fractions? → LCM/HCF of numerators & denominators
• Remainders? → LCM + remainder

Step 2: Break Down the Problem

• For LCM/HCF: List numbers, find prime factors.
• For cyclicity: Find cycle length, compute ( b \mod \text{cycle} ).
• For remainders: Find LCM of divisors, add remainder.

Step 3: Apply the Right Formula/Technique

• LCM/HCF: Use prime factorization or ( \text{LCM} \times \text{HCF} = a \times b ).
• Cyclicity: Use cycle tables.
• Fractions: Apply LCM/HCF rules for fractions.

Step 4: Verify with Answer Choices (MCQ) or Cross-Check (TITA)

• MCQ: Eliminate options that don’t fit (e.g., last digit must be 0–9).
• TITA: Recheck calculations.

Step 5: Optimize for Speed

• Memorize cycles (2,3,4,7,8,9).
• Use shortcuts (e.g., ( \text{LCM}(a, b) = \frac{a \times b}{\text{HCF}(a, b)} )).

Fully Worked CAT-Style Example

Question:
Find the smallest number which when divided by 4, 5, and 6 leaves remainders 3, 4, and 5 respectively.

Solution (Step-by-Step):

1. Identify Problem Type:

2. Remainder problem → Use LCM + remainder adjustment.

3. Understand the Pattern:

4. When divided by 4, remainder = 3 → Number = ( 4k - 1 ).
5. When divided by 5, remainder = 4 → Number = ( 5m - 1 ).
6. When divided by 6, remainder = 5 → Number = ( 6n - 1 ).

7. Common form: Number = LCM(4,5,6) × t - 1.

8. Find LCM of 4,5,6:

9. ( 4 = 2^2 ), ( 5 = 5 ), ( 6 = 2 \times 3 ).

10. LCM = ( 2^2 \times 3 \times 5 = 60 ).

11. Adjust for Remainder:

12. Number = ( 60t - 1 ).

13. Smallest positive number: ( t = 1 ) → ( 60 - 1 = 59 ).

14. Verify:

15. ( 59 \div 4 = 14 ) R3 ✔️
16. ( 59 \div 5 = 11 ) R4 ✔️
17. ( 59 \div 6 = 9 ) R5 ✔️

Answer: 59

Common Mistakes

1. Misapplying LCM/HCF Formula for >2 Numbers

• Mistake: Using ( \text{LCM}(a,b,c) = \frac{a \times b \times c}{\text{HCF}(a,b,c)} ).
• Why it happens: Confusing the two-number formula with three-number cases.
• Correct approach: Use prime factorization for >2 numbers.

2. Ignoring Cyclicity for Last Digit Problems

• Mistake: Calculating ( 7^{2023} ) directly instead of using cycles.
• Why it happens: Overcomplicating the problem.
• Correct approach: Memorize cycles and use ( b \mod \text{cycle} ).

3. Forgetting to Adjust for Remainders

• Mistake: Finding LCM(4,5,6) = 60 and stopping there.
• Why it happens: Not accounting for the remainder condition.
• Correct approach: Subtract 1 (since remainders are 3,4,5 for divisors 4,5,6).

4. Confusing LCM/HCF of Fractions

• Mistake: Taking LCM of denominators instead of HCF.
• Why it happens: Mixing up the rules.
• Correct approach:
• LCM of fractions: ( \frac{\text{LCM(num)}}{\text{HCF(den)}} )
• HCF of fractions: ( \frac{\text{HCF(num)}}{\text{LCM(den)}} )

CAT Traps & Time Management

1. Trap: "Smallest Number" vs. "Largest Number"

• Trap: CAT may ask for the largest number ≤ X with a property (e.g., "largest 3-digit number divisible by 4,5,6").
• How to avoid: Read carefully—"smallest" vs. "largest" changes the approach.

2. Trap: Cyclicity with Negative Remainders

• Trap: ( 2023 \mod 4 = 3 ), but some students take the 4th digit instead of the 3rd.
• How to avoid: Remainder = 0 → last digit of cycle.

3. Time Management

• LCM/HCF (2 numbers): <1 min (use formula).
• LCM/HCF (3+ numbers): 1–2 min (prime factorization).
• Cyclicity: <30 sec (memorize cycles).
• Remainder problems: 1–2 min (LCM + adjustment).

Golden Rule: If stuck for >2 min, move on and return later.

Quick Practice

Question 1:

What is the last digit of ( 3^{100} + 4^{100} )? Answer: 7
Explanation:
- ( 3^{100} ): Cycle of 4 → ( 100 \mod 4 = 0 ) → last digit = 1.
- ( 4^{100} ): Cycle of 2 → ( 100 \mod 2 = 0 ) → last digit = 6.
- 1 + 6 = 7.

Question 2:

Find the HCF of ( \frac{2}{3} ), ( \frac{4}{5} ), and ( \frac{6}{7} ). Answer: ( \frac{2}{105} ) Explanation:
- HCF of numerators (2,4,6) = 2.
- LCM of denominators (3,5,7) = 105.
- HCF = ( \frac{2}{105} ).

Last-Minute Cram Sheet

1. LCM × HCF = a × b (only for two numbers).
2. Cyclicity:
3. 2,3,7,8 → 4
4. 4,9 → 2
5. 5,6 → 1
6. Last digit of ( a^b ): Find ( b \mod \text{cycle} ).
7. LCM of fractions: ( \frac{\text{LCM(num)}}{\text{HCF(den)}} ).
8. HCF of fractions: ( \frac{\text{HCF(num)}}{\text{LCM(den)}} ).
9. Remainder problems: Number = LCM(divisors) × t + remainder.
10. Co-prime numbers: LCM = product, HCF = 1.
11. Smallest number with remainders: LCM - (divisor - remainder).
12. Trap: "Largest number" vs. "smallest number" changes approach.
13. Time: Spend <2 min per question; skip if stuck.

Final Tip:
Practice 50+ questions on LCM/HCF/cyclicity from CAT previous papers & mocks. Speed comes from pattern recognition, not brute force.

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