Common factors and common multiples

When two numbers go on a date, they look for things in common. A common factor is a number that divides both of them. A common multiple is a number that both of them divide. Among all common factors there is a biggest one, and among all common multiples there is a smallest one. These two , the HCF and LCM , solve many everyday puzzles.

Concept

Common factors and HCF

A common factor of $a$ and $b$ is a number that is a factor of both. For $12$ and $18$:

• Factors of $12$: $1, 2, 3, 4, 6, 12$.
• Factors of $18$: $1, 2, 3, 6, 9, 18$.
• Common: $1, 2, 3, 6$.

The highest common factor (HCF) , also called the greatest common divisor (GCD) , is the biggest common factor. For $12$ and $18$, the HCF is $6$.

If the HCF of two numbers is $1$, they are called coprime (or relatively prime). For example, $8$ and $15$ are coprime, even though neither is prime.

Common multiples and LCM

A common multiple of $a$ and $b$ is a number that is a multiple of both. For $4$ and $6$:

• Multiples of $4$: $4, 8, 12, 16, 20, 24, 28, \dots$
• Multiples of $6$: $6, 12, 18, 24, 30, \dots$
• Common: $12, 24, 36, \dots$

The lowest common multiple (LCM) is the smallest one. For $4$ and $6$, the LCM is $12$.

Finding HCF and LCM using prime factorisation

Write each number as a product of primes. Then:

• HCF = product of the minimum power of each common prime.
• LCM = product of the maximum power of every prime appearing in either number.

Example: $18 = 2 \times 3^2$ and $24 = 2^3 \times 3$.

• HCF: $\min(2^1, 2^3) \times \min(3^1, 3^2) = 2^1 \times 3^1 = 6$.
• LCM: $\max(2^1, 2^3) \times \max(3^1, 3^2) = 2^3 \times 3^2 = 72$.

A useful identity:

$\mathrm{HCF}(a, b) \times \mathrm{LCM}(a, b) = a \times b$

Check: $6 \times 72 = 432$ and $18 \times 24 = 432$. ✓

When are HCF and LCM useful?

• HCF problems. "The largest tile that fits exactly into rooms $12$ m and $18$ m wide" , HCF.
• LCM problems. "Two bells ring every $4$ and $6$ seconds. When do they ring together?" , LCM.
• Sharing equally. "Largest number of equal boxes for $12$ apples and $18$ oranges with no fruit mixed" , HCF.
• Repeating events. "Festival every $30$ days and class every $7$ days both fall on" , LCM.

Worked examples

Example 1. Find the HCF of $24$ and $36$.

• $24 = 2^3 \times 3$, $36 = 2^2 \times 3^2$.
• HCF: $2^2 \times 3 = 12$.

Example 2. Find the LCM of $24$ and $36$.

• $24 = 2^3 \times 3$, $36 = 2^2 \times 3^2$.
• LCM: $2^3 \times 3^2 = 72$.
• Check: $\mathrm{HCF} \times \mathrm{LCM} = 12 \times 72 = 864 = 24 \times 36$. ✓

Example 3. Are $9$ and $16$ coprime?

• $9 = 3^2$, $16 = 2^4$. No common prime factor.
• HCF $= 1$. Yes, coprime.

Example 4. Two flashing lights blink every $6$ seconds and every $9$ seconds. They both blink at $t = 0$. When do they next blink together?

• LCM of $6$ and $9$. $6 = 2 \times 3$, $9 = 3^2$. LCM $= 2 \times 3^2 = 18$.
• They blink together at $t = 18$ s.

Example 5. Two tiles of sizes $48$ cm and $60$ cm need to be cut into equal-sized smaller square tiles, as big as possible. What is the largest tile size?

• HCF of $48$ and $60$.
• $48 = 2^4 \times 3$, $60 = 2^2 \times 3 \times 5$.
• HCF $= 2^2 \times 3 = 12$.
• Largest tile: $12$ cm $\times$ $12$ cm.

Try it yourself

1. Find the HCF and LCM of $15$ and $20$.
2. Find the HCF and LCM of $36$ and $48$.
3. Are $14$ and $25$ coprime?
4. Find the smallest number that is a multiple of $4$, $6$, and $9$.
5. Two friends meet every $12$ days and $18$ days. When are they next together?
6. The HCF of two numbers is $7$ and their LCM is $42$. If one is $14$, find the other.
7. Find the largest possible length of a measuring rod that can measure $84$ cm and $112$ cm exactly.
8. Investigate: if $a$ divides $b$, what are HCF$(a, b)$ and LCM$(a, b)$?

Activity

Idli-Vada-like game. Sit in a circle with $5$ or more friends. Pick two numbers, say $3$ and $5$. Going around, each person says the next number in turn, but if the number is a multiple of $3$ they say "idli", a multiple of $5$ they say "vada", and a multiple of both they say "idli-vada". Play till $30$. The "idli-vada" calls happen at multiples of the LCM of $3$ and $5$ , that is, $15$ and $30$. Confirm this works for other pairs too.