LCM , Lowest Common Multiple

The LCM of two numbers is the smallest positive number that both numbers divide.

Idea

Example: LCM of $4$ and $6$. Multiples of $4$: $4, 8, 12, 16, 20, 24, \dots$. Multiples of $6$: $6, 12, 18, 24, \dots$. Common multiples: $12, 24, \dots$. Smallest: $12$. So $\text{LCM}(4, 6) = 12$.

Method 1: Listing multiples. Write multiples until you find a common one. Easy for small numbers.

Method 2: Prime factorisation. Write each as product of primes. For each prime present in any factorisation, take the higher power. Multiply.

Example: $12 = 2^2 \times 3$ and $18 = 2 \times 3^2$. All primes: $2, 3$. Higher powers: $2^2$ and $3^2$. LCM $= 4 \times 9 = 36$.

Method 3: Using the HCF–LCM identity. For two numbers $a$ and $b$: $\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b.$ So if you know the HCF, the LCM is $\dfrac{a \times b}{\text{HCF}(a, b)}$.

Example: $a = 12, b = 18, \text{HCF} = 6$. So $\text{LCM} = \tfrac{12 \times 18}{6} = 36$. ✓

Important: the HCF–LCM identity works only for two numbers. For three or more, use prime factorisation.

Where LCM is used.

• Adding fractions: common denominator is the LCM.
• Scheduling: when two events repeat at intervals $a$ and $b$, the next coincidence is after $\text{LCM}(a, b)$.
• Cyclic problems: lights blinking, traffic lights syncing.

Worked examples

Example 1. $\text{LCM}(6, 8)$ by listing.

Multiples of $6$: $6, 12, 18, 24, \dots$. Multiples of $8$: $8, 16, 24, \dots$. Common: $24, 48, \dots$. Smallest: $24$.

Example 2. $\text{LCM}(24, 36)$ by prime factorisation.

$24 = 2^3 \times 3$ and $36 = 2^2 \times 3^2$. Higher powers: $2^3$ and $3^2$. LCM $= 8 \times 9 = 72$.

Example 3. $\text{LCM}(12, 18)$ using HCF.

$\text{HCF}(12, 18) = 6$. So $\text{LCM} = \tfrac{12 \times 18}{6} = 36$.

Example 4. Two buses leave a stop every $15$ and $20$ minutes. When do they next leave together?

$\text{LCM}(15, 20) = 60$ minutes , they leave together every hour.

Try it yourself

1. $\text{LCM}(4, 5)$.
2. $\text{LCM}(8, 12)$ by prime factorisation.
3. $\text{LCM}(15, 25)$.
4. $\text{LCM}$ of $a = 24, b = 36$ using $\text{HCF}=12$.
5. $\text{LCM}(6, 10, 15)$.
6. Two cyclists round a track in $40$ s and $60$ s respectively. Time to next meet at start?
7. $\text{LCM}(7, 11)$ , explain.
8. Find LCM of $9, 12, 16$.

Activity

Make a "circle of multiples": draw concentric ovals labelled with multiples of $4$ and $6$. The intersections are the common multiples. The innermost intersection is the LCM.