LCM and HCF

Module 02 — Lesson 0002 · In Depth · 40 min

Factors divide, multiples multiply. HCF is the biggest divisor common to all numbers; LCM is the smallest dividend common to all. Prime factorization is the reliable engine; a×b = HCF×LCM (for two numbers only) and the h·x, h·y co-prime model are the shortcuts.

1. Basic Definitions

Term	Definition	Example
Factors	Numbers that divide exactly (remainder 0)	Factors of 12: 1,2,3,4,6,12
Multiples	Numbers obtained by multiplying by an integer	Multiples of 4: 4,8,12,16,20…
HCF / GCD	Largest number dividing two or more numbers	HCF(12,18)=6
LCM	Smallest number divisible by two or more numbers	LCM(12,18)=36
Co-prime	HCF is 1 — share no factor other than 1	8 and 9 are co-prime (neither is prime)

2. Prime Factorization Method — The Reliable Way

Step 1: Prime Factorization

Break each number into primes raised to powers.

Example: 12 and 18 → 12 = 2² × 3¹, 18 = 2¹ × 3²

Step 2: HCF — Lowest Power of Common Primes

Common primes 2 and 3 → lowest: 2¹, 3¹ → HCF = 2¹×3¹=6

Step 3: LCM — Highest Power of Every Prime Involved

Highest: 2², 3² → LCM = 2²×3²=4×9=36

3. The Golden Rule — For Two Numbers Only

a × b = HCF(a,b) × LCM(a,b)

Usage: given HCF, LCM, and one number → second = HCF×LCM / first.

Warning: does NOT work for three or more numbers — a×b×c ≠ HCF×LCM in general.

Worked: HCF=6, LCM=36, a=12 → b=6×36/12=18 ✓.

4. Advanced Algebraic Model — The h·x, h·y Model

When numbers are unknown (e.g., "sum is 96"), let h = HCF, represent numbers as:

First = h·x, Second = h·y — with crucial condition: x and y must be co-prime (otherwise h would be larger).

Derived Formula	Expression
Sum	`h(x+y)`
Difference	`h(x−y)`
Product	`h²xy`
LCM	`hxy` (because common factor h plus co-prime x,y)

Application: "LCM is 15×HCF" → hxy=15h → xy=15. Find co-prime pairs with product 15: (1,15) and (3,5) — both co-prime. So if h=6, numbers are 6&90 or 18&30. The non-co-prime pair (5,3) is same as (3,5) — order doesn't matter.
Sum example: Sum=96, HCF=8 → 8(x+y)=96 → x+y=12. Co-prime pairs summing to 12: (1,11), (5,7) — (3,9) rejected (HCF 3), (2,10) rejected.

5. Remainder Models (Word Problems)

Type A: Finding the Dividend — Smallest Number / LCM Based

"Find smallest number which when divided by X, Y, Z …"

Condition	Formula
Leaves no remainder (exactly divisible)	`LCM(X,Y,Z)`
Leaves same remainder `r` in each case	`LCM(X,Y,Z) + r`
Leaves different remainders, but `Divisor − Remainder = k` constant	`LCM(X,Y,Z) − k`

Worked k-constant: Divided by 3 leaves 1, by 4 leaves 2, by 5 leaves 3 → each D−R=2 → answer = LCM(3,4,5)−2=60−2=58. Check: 58/3=19 r1 ✓, /4 r2 ✓.

Type B: Finding the Divisor — Largest Number / HCF Based

"Find largest number which divides A, B, C …"

Condition	Formula
Leaves remainders `p,q,r`	`HCF(A−p, B−q, C−r)` — subtract remainders first
Leaves same remainder (value unknown)	`HCF(\|A−B\|, \|B−C\|, \|C−A\|)` — differences

Worked: Largest number which divides 42, 63, 84 leaving same remainder → differences 21,21,42 → HCF=21 → check: 42 mod21=0, 63 mod21=0, 84 mod21=0 → remainder 0 works; any divisor of 21 also works but 21 is largest.

6. Properties and Miscellaneous

HCF divides LCM: LCM is multiple of HCF. If someone says HCF=12, LCM=360 → valid (360%12=0). If LCM=350 with HCF=12 → impossible.
Counting pairs: "How many pairs with sum S and HCF h?" → set x+y=S/h, count co-prime pairs summing to that — only co-prime pairs count.
Fractions: HCF of fractions = HCF(numerators)/LCM(denominators), LCM of fractions = LCM(numerators)/HCF(denominators) — note inversion.
Co-prime / prime numbers: If numbers are co-prime, HCF=1 and LCM = product (since no common factor). If they are prime, same.

7. Summary of Variables

Symbol	Meaning
`N`	Number sought
`h / l`	HCF / LCM
`a,b,c`	Given numbers (divisors/dividends)
`x,y`	Co-prime factors: `a=hx, b=hy`
`r / k`	Remainder / constant difference

HCF(12,18) and LCM(12,18)?

a×b = ?

Does a×b=HCF×LCM work for 3 numbers?

If h=6 and xy=15, possible numbers?

Smallest number divisible by 3,4,5 with no remainder?

Divided by 3 leaves1, by4 leaves2, by5 leaves3 (D−R=2 constant) → smallest?

Largest number dividing 42,63,84 leaving same remainder → ?

HCF of fractions a/b and c/d = ?

Notes

Basics :-

Factors ⇒ divide exactly ; Multiples ⇒ n×integer

HCF ⇒ largest common divisor ; LCM ⇒ smallest common multiple

Co-prime ⇒ HCF=1 (eg 8,9)

Prime Factorization :-

12=2²×3, 18=2×3² → HCF=2×3=6 (lowest), LCM=2²×3²=36 (highest)

Golden Rule :-

a×b = HCF×LCM — only for 2 numbers

h·x Model :-

Numbers ⇒ hx, hy where x,y co-prime ; sum h(x+y), diff h(x−y), product h²xy, LCM hxy

(eg h=6, xy=15 → (3,5)→18,30)

Remainders :-

Type A (LCM): no remainder→LCM ; same r→LCM+r ; D−R=k→LCM−k

Type B (HCF): remainders p,q,r→HCF(A−p,…) ; same unknown→HCF of differences

Properties :-

HCF divides LCM ; co-prime → LCM=product ; fractions: HCF=HCFnum/LCMden, LCM=LCMnum/HCFden

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