HCF & LCM: Fastest Tricks & Complete Guide for Bank SSC Railway Exams

What are HCF and LCM?

HCF (Highest Common Factor) — also called GCD (Greatest Common Divisor) — is the largest number that divides two or more numbers without leaving any remainder.

LCM (Least Common Multiple) is the smallest number that is exactly divisible by two or more numbers.

In competitive exams, HCF and LCM appear in 3 forms: (1) Direct calculation, (2) Application-based word problems (tiles, bells, ropes, containers), and (3) Number theory based (find numbers given HCF/LCM).

Key Relation: HCF × LCM = Product of the two numbers
(Valid for exactly 2 numbers only)

Method 1: Prime Factorization

Break each number into its prime factors. Then:

✅ HCF = Product of common prime factors with lowest power
✅ LCM = Product of all prime factors with highest power

🧮 Example 1: Find HCF and LCM of 36 and 48

36 = 2² × 3²
48 = 2⁴ × 3

HCF = 2² × 3¹ = 4 × 3 = 12
LCM = 2⁴ × 3² = 16 × 9 = 144

Verification: 12 × 144 = 1728 = 36 × 48 ✅

🧮 Example 2: Find HCF and LCM of 12, 18, 24

12 = 2² × 3
18 = 2 × 3²
24 = 2³ × 3

HCF = 2¹ × 3¹ = 6
LCM = 2³ × 3² = 8 × 9 = 72

Method 2: Division Method (Faster for 2 numbers)

For finding HCF of two large numbers, the Division Method (Euclidean Algorithm) is the fastest approach in exams:

Divide the larger number by the smaller.
Divide the divisor by the remainder.
Repeat until remainder = 0.
The last divisor is the HCF.

🧮 Example 3: Find HCF of 252 and 105

252 ÷ 105 = 2 remainder 42
105 ÷ 42 = 2 remainder 21
42 ÷ 21 = 2 remainder 0

HCF = 21 ✅
LCM = (252 × 105) / 21 = 26460 / 21 = 1260

Type 1: Bell / Timer Problems (LCM Application)

This is the most classic HCF/LCM application problem type — two or more events repeat at regular intervals, and you need to find when they next occur simultaneously.

Key Insight: When will multiple events happen together again = LCM of their individual intervals.

🧮 Example 4: Three bells ring at intervals of 12, 18, and 24 minutes. If they all ring together at 8:00 AM, at what time will they next ring together?

LCM of 12, 18, 24 = 72 minutes
Next time = 8:00 AM + 72 minutes = 9:12 AM ✅

🧮 Example 5: Three athletes complete a lap in 8, 12, and 16 minutes respectively. If they start together, after how many minutes will they be at the starting point simultaneously?

LCM of 8, 12, 16 = 48 minutes
Answer: 48 minutes ✅

Type 2: Tile / Container Problems (HCF Application)

When you need to find the largest size of something that fits perfectly — that's always HCF!

Key Insight: Largest possible size = HCF of the given dimensions.

🧮 Example 6: What is the largest square tile that can exactly cover a floor of 180 cm × 120 cm without cutting?

HCF of 180 and 120:
180 = 2² × 3² × 5
120 = 2³ × 3 × 5
HCF = 2² × 3 × 5 = 60 cm ✅

🧮 Example 7: Largest container that can measure 420 litres, 630 litres, and 840 litres exactly?

HCF of 420, 630, 840 = 210 litres ✅

Type 3: Find Numbers Given HCF & LCM

These are reverse problems — HCF and LCM are given, and you need to find the original numbers. These appear frequently in SSC CGL and IBPS PO Mains.

Key Property: If HCF = h, then the two numbers can be written as ha and hb where a and b are co-prime (HCF of a, b = 1). Then LCM = h × a × b.

🧮 Example 8: HCF of two numbers is 12 and their LCM is 144. One number is 36. Find the other.

Other number = (HCF × LCM) / First number = (12 × 144) / 36 = 1728 / 36 = 48 ✅

🧮 Example 9: The HCF and LCM of two numbers are 8 and 96. If one number is 24, find the other.

Other = (8 × 96) / 24 = 768 / 24 = 32 ✅

💡 Exam Trick: The two numbers are ALWAYS factors of LCM AND multiples of HCF. So to quickly verify your answer: check if both numbers are divisible by the HCF AND if the LCM is divisible by both numbers.

Type 4: Remainder-Based Problems

These advanced questions say: "Find the smallest number which when divided by a, b, c leaves remainder r."

✅ Same remainder r: Answer = LCM(a, b, c) + r
✅ Deficient by same amount d: Answer = LCM(a, b, c) – d
✅ Different remainders: Find the "deficiency" = divisor – remainder; if all deficiencies are equal → Answer = LCM – deficiency

🧮 Example 10: Find the smallest number divisible by 6, 9, 12 that leaves remainder 5 in each case.

LCM of 6, 9, 12 = 36
Answer = 36 + 5 = 41 ✅

🧮 Example 11: Find the smallest number which when divided by 5, 6, 7 leaves remainders 3, 4, 5 respectively.

Deficiencies: 5–3=2, 6–4=2, 7–5=2 (all same!)
LCM of 5, 6, 7 = 210
Answer = 210 – 2 = 208 ✅

📋 Quick Revision Table: HCF & LCM at a Glance

Problem Type	Use	Formula
Tiles, largest container	HCF	Largest size that fits exactly
Bells, athletes meet	LCM	Next time events coincide
Find other number	Relation	HCF × LCM / known number
Same remainder r	LCM	LCM + r
Deficiency same (d)	LCM	LCM – d

Practice Questions: HCF & LCM (With Answers)

Solve these questions using the methods you learned above:

Find HCF of 84 and 126.
Answer: 84=2²×3×7, 126=2×3²×7. HCF=2×3×7=42
Find LCM of 15, 25, and 35.
Answer: 15=3×5, 25=5², 35=5×7. LCM=3×5²×7=525
HCF of two numbers is 15 and LCM is 300. One number is 60. Find the other.
Answer: (15×300)/60=75
Find the smallest number which when divided by 12, 18, 24 leaves remainder 7 in each case.
Answer: LCM(12,18,24)=72. Answer=72+7=79
Three bells ring at intervals of 24, 36, and 48 minutes. They ring together at 9:00 AM. When next?
Answer: LCM=144 min = 2 hrs 24 min. Next at 11:24 AM

Previous Year Questions: HCF & LCM in Exams

🏦 SBI Clerk 2023: The product of two numbers is 2028 and their HCF is 13. Find the number of such pairs.
Solution: LCM = 2028/13 = 156 = 4×39 = 12×13. Pairs where HCF=13: (13,156), (52,39). Answer: 2 pairs ✅

🚂 Railway NTPC 2021: What is the largest number that divides 25, 73, and 97 leaving remainders 5, 3, and 7 respectively?
Solution: Reduce: 25–5=20, 73–3=70, 97–7=90. HCF(20,70,90)=10. Answer: 10 ✅

🏦 IBPS PO 2022: Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10, 12 seconds respectively. In 30 minutes, how many times do they toll together?
Solution: LCM(2,4,6,8,10,12)=120 seconds=2 minutes. 30/2=15 times. Plus 1 (initial)=16 times ✅

Advanced HCF & LCM Concepts for IBPS PO & SSC CGL Mains

These advanced concepts appear in Mains-level papers and require a deeper understanding beyond basic HCF/LCM calculation:

Co-prime Numbers

Two numbers are co-prime if their HCF = 1. When HCF of two numbers is h, those two numbers can be written as h×a and h×b where a and b are co-prime to each other. This concept is critical in "how many such pairs are possible" type questions.

Example: HCF = 12, LCM = 120. Numbers = 12a and 12b where ab = 120/12 = 10. Co-prime pairs (a,b) where a×b=10: (1,10) and (2,5). So 2 pairs of numbers are possible: (12,120) and (24,60).

LCM for Fractions

LCM of Fractions = LCM of Numerators / HCF of Denominators
HCF of Fractions = HCF of Numerators / LCM of Denominators

Example: LCM of 1/2, 3/4, 5/6 = LCM(1,3,5) / HCF(2,4,6) = 15/2

Example: HCF of 2/3, 4/9, 8/27 = HCF(2,4,8) / LCM(3,9,27) = 2/27

Important Properties to Remember

✅ HCF always divides LCM exactly (LCM is always a multiple of HCF)
✅ For prime numbers a and b: HCF(a,b) = 1, LCM(a,b) = a×b
✅ If one number is a multiple of another: HCF = smaller number, LCM = larger number
✅ HCF of (a, b, c) always divides LCM of (a, b, c)
✅ HCF × LCM = Product is ONLY valid for exactly 2 numbers, NOT for 3 or more

⚡ Speed Tricks: Solve HCF & LCM in Under 60 Seconds

Top rankers use these mental shortcuts to never spend more than 60 seconds on an HCF/LCM question:

✅ HCF of consecutive numbers: HCF(n, n+1) = 1 always (consecutive numbers are always co-prime)
✅ LCM of consecutive numbers: LCM(n, n+1) = n×(n+1) always
✅ Quick HCF using subtraction: HCF(a,b) = HCF(a, b–a) for b > a. Useful when numbers are close. E.g., HCF(98,112) = HCF(98,14) = HCF(14,0) = 14
✅ If one number divides the other: HCF = smaller, LCM = larger. No calculation needed!
✅ Same remainder r: HCF of differences = HCF of (a–r), (b–r), (c–r)

🔢 HCF & LCM: Fastest Tricks & Complete Guide for Bank, SSC & Railway Exams