📘 Factors, Multiples, HCF and LCM
This page explains four important number ideas you will use a lot in maths.
🔹 Factors
Numbers: 12 and 18
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
Common factors: 1, 2, 3, 6
✅ HCF = 6
🔹 Multiples
Numbers: 12 and 18
Multiples of 12: 12, 24, 36, 48, 60, 72, ...
Multiples of 18: 18, 36, 54, 72, 90, ...
Common multiples: 36, 72
✅ LCM = 36
🔢 Factors
A factor is a number that divides another number exactly, with no remainder.
Factors of 12 are: 1, 2, 3, 4, 6, 12
(Because 12 ÷ each of these gives a whole number)
1️⃣ Start with 1 and divide the number by it.
2️⃣ Write down both numbers if it divides exactly.
3️⃣ Keep going up to the square root of the number.
4️⃣ Stop when the numbers start repeating.
24 ÷ 1 = 24 → factors: 1 and 24
24 ÷ 2 = 12 → factors: 2 and 12
24 ÷ 3 = 8 → factors: 3 and 8
24 ÷ 4 = 6 → factors: 4 and 6
(After this, the pairs repeat, so we stop.)
All factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Every number is a factor of itself
🔁 Multiples
A multiple is the result of multiplying a number by a whole number.
Multiples of 5 are: 5, 10, 15, 20, 25, ...
- Multiples go on forever
- A number can have many multiples
🏆 HCF (Highest Common Factor) or GCD (Greatest Common Divisor)
The HCF is the largest factor that two or more numbers share.
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Common factors: 1, 2, 3, 6
HCF = 6
📦 What is HCF? (With Examples)
HCF means Highest Common Factor. It helps us when we want to share things fairly or split things into equal groups with nothing left over.
🧠 Think like this:
“What is the largest number that can divide everything exactly?”
📖 Example 1: Sharing pens
A teacher has 12 red pens and 18 blue pens. She wants to give the pens to students so that:
- Each student gets the same number of red pens
- Each student gets the same number of blue pens
- No pens are left over
🔍 Step 1: What are we doing?
We are sharing two different things fairly. So we need HCF.
🔢 Step 2: Find the HCF
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
The largest factor they both share is 6.
✅ So, the HCF of 12 and 18 is 6.
Each student gets:
12 ÷ 6 = 2 red pens
18 ÷ 6 = 3 blue pens
📖 Example 2: Making equal boxes
A baker has 20 cookies and 30 cakes. He wants to pack them into the largest number of identical boxes with no food left over.
🔍 Step 1: What is being asked?
We are making the largest number of equal groups. That tells us to use HCF.
🔢 Step 2: Find the HCF
HCF(20, 30) = 10
✅ The baker can make 10 identical boxes.
Each box will have:
20 ÷ 10 = 2 cookies
30 ÷ 10 = 3 cakes
⭐ Remember:
Use HCF when you are
sharing different things fairly
into the largest number of equal groups,
with nothing left over.
🔗 LCM (Lowest Common Multiple)
The LCM is the smallest multiple that two or more numbers have in common.
Multiples of 4: 4, 8, 12, 16, 20, 24...
Multiples of 6: 6, 12, 18, 24...
Common multiples: 12, 24...
LCM = 12
📖 Example: When do we use LCM?
A school bell rings every 4 minutes. Another bell rings every 6 minutes.
Both bells ring at the same time now. When will they ring together again?
🔍 Step 1: What is happening?
- The bells ring again and again
- We are looking for the next time they ring together
- Nothing is being shared or divided
🔢 Step 2: List the multiples
Multiples of 4:
4, 8, 12, 16, 20, …
Multiples of 6:
6, 12, 18, 24, …
⭐ Step 3: Find the first number they share
The first number that appears in both lists is 12.
✅ So, the bells will ring together again after 12 minutes.
⭐ Remember: Use LCM when things repeat and you want to know when they match up again.
🧠 Quick Summary
- Factors divide exactly
- Multiples are results of multiplication
- HCF = biggest shared factor
- LCM = smallest shared multiple
- In school maths: Factors, Multiples, HCF and LCM are always positive