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LCM & HCF — The Two Numbers That Rule Every Exam

The Hook: Two Buses, One Question

Two buses leave a depot together at 6:00 AM. Bus A returns every 12 minutes, Bus B every 18 minutes. When will they be at the depot together again?

You could list times on paper for 10 minutes… or you could find the LCM of 12 and 18 — which is 36 — and answer in 10 seconds: 6:36 AM.

Now flip the question: A fruit seller has 48 apples and 60 oranges. He wants identical baskets with no fruit left over. What is the maximum number of baskets? That's the HCF of 48 and 60 — which is 12.

Same two numbers. Two different questions. Two different answers. That single difference — "together again" vs. "maximum equal groups" — is the heart of this chapter, and questions built on it appear regularly in SSC, Banking, and Railway exams.

Why Students Struggle (And Why You Won't)

Most students don't fail LCM & HCF because the maths is hard. They fail because:

1. They mix up when to use LCM and when to use HCF.
2. They forget the golden formula connecting the two.
3. They panic when remainders enter the picture ("leaves remainder 3 in each case…").

We will fix all three, one small chunk at a time. Read slowly. Pause at every 🧠 prompt and actually answer it before moving on — that single habit doubles retention.

Chunk 1: What Do These Words Even Mean?

Multiple — what you get when you multiply a number. Multiples of 4: 4, 8, 12, 16, 20, 24… (they go on forever, getting bigger).

Factor (Divisor) — what divides a number exactly. Factors of 12: 1, 2, 3, 4, 6, 12 (a finite, small list).

Now the two stars:

• LCM (Least Common Multiple) = the *smallest* number that is a multiple of *all* the given numbers. Think: a common meeting point in the future.
• HCF (Highest Common Factor) = the *largest* number that divides *all* the given numbers exactly. Also called GCD (Greatest Common Divisor).

🧠 Active Recall: Without looking back — which one is bigger than the original numbers, LCM or HCF? Which list is endless: multiples or factors?

Memory anchor: LCM = Large Common Multiple (usually ≥ the numbers). HCF = Humble Common Factor (usually ≤ the numbers). "LCM is Large, HCF is Humble."

Chunk 2: Method 1 — Prime Factorization (The Reliable Workhorse)

Break every number into primes, then:

• HCF: take only the common primes, each raised to the lowest power.
• LCM: take all primes that appear, each raised to the highest power.

Example: Find HCF and LCM of 24 and 36.

• 24 = 2³ × 3¹
• 36 = 2² × 3²

HCF = 2² × 3¹ = 12 (common primes, lowest powers)
LCM = 2³ × 3² = 72 (all primes, highest powers)

🧠 Active Recall: For 18 = 2 × 3² and 30 = 2 × 3 × 5, write the HCF and LCM before reading on. (HCF = 6, LCM = 90.)

Chunk 3: Method 2 — Division Method (For Big Numbers)

Prime factorization gets painful with numbers like 391 and 437. Use the Euclidean division method for HCF:

1. Divide the bigger number by the smaller.
2. Replace the bigger number with the smaller one, and the smaller one with the remainder.
3. Repeat until the remainder is 0. The last non-zero divisor is the HCF.

Example: HCF of 391 and 437.

• 437 ÷ 391 → remainder 46
• 391 ÷ 46 → remainder 23 (46 × 8 = 368; 391 − 368 = 23)
• 46 ÷ 23 → remainder 0

HCF = 23. Then use the golden formula (next chunk) to get the LCM.

🧠 Active Recall: Why does the division method beat factorization for large numbers? Say the reason out loud.

Chunk 4: The Golden Formula

For any two numbers a and b:

> LCM × HCF = a × b

This is the single most-used identity in this chapter. Given any three of the four quantities, you get the fourth instantly.

Example: HCF of two numbers is 11, LCM is 693. One number is 77. Find the other.
Other number = (11 × 693) / 77 = 99.

⚠️ Warning: This formula works for two numbers only — not three. Examiners love trapping students here.

🧠 Active Recall: Two numbers have HCF 4 and LCM 48. Their product is…? (Answer: 192.)

Chunk 5: LCM & HCF of Fractions

Memorize this pair — it appears directly in exams:

> LCM of fractions = LCM of numerators / HCF of denominators
> HCF of fractions = HCF of numerators / LCM of denominators

Example: LCM of 2/3, 4/5, 8/15 = LCM(2, 4, 8) / HCF(3, 5, 15) = 8/1 = 8.

Mnemonic (the weird one): 🍕 *"LCM is a greedy giant — he EATS the top (LCM of numerators) and CRUSHES the bottom (HCF of denominators). HCF is his polite twin who does the exact opposite."* Picture a giant munching numerators. You'll never swap them again.

🧠 Active Recall: HCF of 3/4 and 9/8 = ? (HCF(3,9)/LCM(4,8) = 3/8.)

Chunk 6: The Remainder Family (Where Exams Are Won)

These four templates cover most "tricky" questions:

Type 1 — Greatest number dividing a, b, c leaving the SAME remainder r:
HCF of (a−r), (b−r), (c−r).

Type 2 — Greatest number dividing a, b, c leaving DIFFERENT remainders r₁, r₂, r₃:
HCF of (a−r₁), (b−r₂), (c−r₃).

Type 3 — Smallest number which, when divided by a, b, c, leaves the SAME remainder r:
LCM(a, b, c) + r.

Type 4 — Smallest number which, when divided by a, b, c, leaves remainders such that (divisor − remainder) is the SAME constant k:
LCM(a, b, c) − k.

Example (Type 4): Smallest number which when divided by 4, 5, 6 leaves remainders 1, 2, 3 respectively. Here k = 3 in each case (4−1 = 5−2 = 6−3 = 3). Answer = LCM(4,5,6) − 3 = 60 − 3 = 57.

🧠 Active Recall: Cover the table above. Which type uses "LCM + r"? Which uses "LCM − k"? What is the first step in Type 1?

Chunk 7: Co-Primes and the "Ratio" Shortcut

Co-prime numbers share no common factor except 1 (e.g., 8 and 15). For co-primes: HCF = 1 and LCM = their product.

Ratio shortcut: If the HCF of two numbers is H and the numbers are in ratio m : n (where m, n are co-prime), then the numbers are Hm and Hn, and LCM = H × m × n.

Example: Two numbers are in ratio 3 : 4 and their HCF is 5. Numbers = 15 and 20; LCM = 5 × 3 × 4 = 60.

🧠 Active Recall: Are 14 and 21 co-prime? Why not? (Common factor 7.)

Chunk 8: Word-Problem Radar — LCM or HCF?

Train your instinct with this table:

Classic: Find the least number of square tiles to pave a courtyard 15 m 17 cm × 9 m 2 cm. Convert to cm: 1517 × 902. Side of tile = HCF(1517, 902) = 41 cm. Number of tiles = (1517 × 902)/(41 × 41) = 37 × 22 = 814.

🧠 Active Recall: Three bells ring at intervals of 4, 6, 8 seconds. LCM or HCF? (LCM = 24 seconds.)

Summary Table — Pin This in Your Memory

Your Next Step

📌 Review this chapter after 2 days. When you do, don't re-read — instead, try to rewrite the summary table from memory, then check. Whatever you forgot is exactly what your next practice set will fix.

You've just built the complete LCM & HCF toolkit. Now go make those two buses meet. 🚌🚌