HCF vs LCM
HCF (GCD) is the largest integer dividing all numbers; LCM is the smallest positive integer divisible by all numbers.
Intersection vs Union Model
In prime factorized form $A = 2^3 \cdot 3^1$ and $B = 2^2 \cdot 3^3$:
- • $\text{HCF}$: Take MINIMUM power of shared prime factors $\implies 2^2 \cdot 3^1 = \mathbf{12}$.
- • $\text{LCM}$: Take MAXIMUM power of all prime factors $\implies 2^3 \cdot 3^3 = \mathbf{216}$.
- • Ratio Representation: If numbers are in ratio $x : y$ with HCF $h$, numbers = $hx, hy$, and $\text{LCM} = hxy$.
HCF & LCM form the foundation for Time & Work cycle rates, circular track meeting times, bell-ringing questions, and fraction simplification:
Product Law (2 Numbers)
If $a = hx, b = hy$, then $\text{LCM} = hxy$ and $h \cdot hxy = (hx)(hy)$.
HCF of Fractions
Example: $\text{HCF}(2/3, 8/9) = \frac{\text{HCF}(2,8)}{\text{LCM}(3,9)} = \frac{2}{9}$.
LCM of Fractions
Example: $\text{LCM}(2/3, 8/9) = \frac{\text{LCM}(2,8)}{\text{HCF}(3,9)} = \frac{8}{3}$.
The 4 Standard Word Problem Remainder Models
Model 1: Largest Divisor (Same Remainder r)
$\text{Required Number} = \text{HCF}(A - r, B - r, C - r)$Model 2: Smallest Dividend (Same Remainder r)
$\text{Required Number} = \text{LCM}(A, B, C) + r$Model 3: Smallest Dividend (Diff Remainders, Constant Diff K)
$\text{If } A - r_1 = B - r_2 = C - r_3 = K \implies \text{LCM}(A, B, C) - K$Model 4: Bell Ringing / Track Meetings
$\text{Interval} = \text{LCM}(\text{individual time periods})$SHORTCUT RULES & SPEED HACKS
Ratio & HCF/LCM Relationship
Since $x$ and $y$ are co-prime ($\gcd(x,y)=1$), their product with HCF gives LCM directly without calculating $A$ and $B$!
Minimum Tiles for Room ($L \times B$)
Finding the fewest number of square tiles needed to pave a rectangular area.
SOLVED EXAMPLES (LEVEL 0 TO HARD)
Q: The HCF and LCM of two numbers are 12 and 240 respectively. If one number is 48, find the other.
Step 1: Apply Product Law: $\text{HCF} \times \text{LCM} = A \times B$.
Step 2: $12 \times 240 = 48 \times B$.
Step 3: $B = \frac{12 \times 240}{48} = \frac{240}{4} = \mathbf{60}$.
Q: Four bells toll together at 9:00 AM. They toll at intervals of 6, 8, 12, and 18 seconds respectively. How many times will they toll together in 1 hour?
Step 1: Find $\text{LCM}(6, 8, 12, 18) = 72$ seconds.
Step 2: Total seconds in 1 hour = $3600$ seconds.
Step 3: Count of tollings = $\lfloor 3600 / 72 \rfloor + 1 = 50 + 1 = \mathbf{51}$ times (including the initial toll at 9:00 AM!).
Q: Find the smallest 4-digit number which when divided by 12, 15, 20, and 35 leaves remainders 8, 11, 16, and 31 respectively.
Step 1: Constant difference $K = 12-8 = 15-11 = 20-16 = 35-31 = \mathbf{4}$.
Step 2: General form $= \text{LCM}(12, 15, 20, 35) \cdot m - 4 = 420m - 4$.
Step 3: Smallest 4-digit number $\ge 1000$: For $m=3 \implies 420(3) - 4 = 1260 - 4 = \mathbf{1256}$.
Mistake 1
Product Law on 3 Numbers
Applying $\text{HCF} \times \text{LCM} = A \times B \times C$. This rule ONLY works for TWO numbers!
Mistake 2
Forgetting Initial Toll (+1)
In bell-ringing questions starting at time zero, forgetting to add $+1$ for the starting simultaneous toll.
Mistake 3
Adding vs Subtracting K
In Model 3 (different remainders with constant difference $K$), subtracting $K$ instead of adding: $\text{LCM} - K$, NOT $\text{LCM} + K$.
1. Find the HCF and LCM of 4/5, 6/25, and 8/15.
Answer: $\text{HCF} = \frac{\text{HCF}(4,6,8)}{\text{LCM}(5,25,15)} = \frac{2}{75}$. $\text{LCM} = \frac{\text{LCM}(4,6,8)}{\text{HCF}(5,25,15)} = \frac{24}{5}$.
2. Two numbers are in ratio 4 : 5. If their HCF is 7, find their LCM.
Answer: $\text{LCM} = 7 \times 4 \times 5 = 140$. Numbers are 28 and 35.
3. Find the largest 3-digit number which when divided by 6 and 9 leaves a remainder of 4.
Answer: $\text{LCM}(6,9) \cdot m + 4 = 18m + 4$. Largest 3-digit multiple of 18 is $990$. Number $= 990 + 4 = 994$.
FREQUENTLY ASKED QUESTIONS
❓ What is the Product Theorem of HCF and LCM?
For any two positive integers a and b: HCF(a, b) * LCM(a, b) = a * b. Note: This identity holds strictly for TWO numbers, not three.
❓ How do you calculate HCF and LCM of fractions?
HCF of fractions = HCF(Numerators) / LCM(Denominators). LCM of fractions = LCM(Numerators) / HCF(Denominators).
❓ How do you solve bell ringing and circular track meeting problems?
Find the LCM of the individual time intervals. For bells ringing every 12, 18, and 24 minutes, they will ring together every LCM(12, 18, 24) = 72 minutes.