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Numbers come in many flavours. Some line up as neat squares, others stack like triangles, some can only be divided by 1 and themselves. In this unit we explore square numbers, triangular numbers, divisibility rules, prime and composite numbers, prime factorisation, LCM and HCF.
Main skills from this lesson
Welcome to Unit 2! Numbers can be sorted into special families: square numbers, triangular numbers, prime numbers, composite numbers and more.
You will also learn shortcuts called divisibility rules and tools like LCM and HCF.
Math mindset: If I don't get it right the first time, I can try a new strategy.
Remember from earlier grades: a multiple of a number is what you get when you multiply that number by 1, 2, 3, 4...
For example, the first five multiples of 2 are 2, 4, 6, 8, 10.
Multiples of 6: 6, 12, 18, 24, 30...
What is the 8th multiple of 2?
Look at a multiplication grid. The numbers on the diagonal - 1, 4, 9, 16, 25, 36... - appear when a number is multiplied by itself.
These are called square numbers.
1, 4, 9, 16, 25, 36, 49, 64, 81, 100...
Why are these called square numbers?
Imagine a box of blocks. A square has the same number of blocks on each side.
The numbers 1, 4, 9, 16 are square numbers.
n x n = n^2
Math Language: When a number is multiplied by itself, the product is called the square of that number.
How many blocks are needed for a 6 x 6 square?
Triangular numbers can be drawn as dots arranged in an equilateral triangle.
Start with 1 dot. Add a row of 2 to get 3. Add a row of 3 to get 6. Add a row of 4 to get 10.
The sequence is 1, 3, 6, 10, 15, 21, 28...
1, 3, 6, 10, 15, 21...
Math Language: Triangular numbers are dots arranged in an equilateral triangle.
What is the 5th triangular number?
The triangular-dot pattern starts 1, 3, 6,....
Each new diagram adds one more dot to the next row, so the next totals are 10, 15, 21.
After 1, 3, 6 - what is the next triangular number?
Complete the square-number table by multiplying each number by itself.
Notice the alternating pattern: odd x odd is odd, even x even is even.
| Calculation | Square number |
|---|---|
| 1 x 1 | 1 |
| 2 x 2 | 4 |
| 3 x 3 | 9 |
| 4 x 4 | 16 |
| 5 x 5 | 25 |
| 6 x 6 | 36 |
| 7 x 7 | 49 |
| 8 x 8 | 64 |
| 9 x 9 | 81 |
| 10 x 10 | 100 |
Which of these square numbers are even?
From the list 1, 3, 35, 45, 17, 15, 36, 10, 81, 55, 121, 28, 144, 100, 25, 21, 123, 89, 6 - pick the triangular numbers.
The triangular numbers are 1, 3, 10, 6, 15, 28, 36, 45, 55, 100, 21.
Which of these is NOT a triangular number?
Match each multiplication to its square number.
2 x 2 = 4, 3 x 3 = 9, 4 x 4 = 16, 5 x 5 = 25, 6 x 6 = 36, 7 x 7 = 49, 8 x 8 = 64, 9 x 9 = 81, 11 x 11 = 121, 12 x 12 = 144.
What is 11 x 11?
If a number can be divided by another number with no remainder, we say the first is divisible by the second.
8 beads can be arranged in 2 equal rows of 4, but 9 beads cannot - 9 is not divisible by 2.
Which number is divisible by 2?
Look at the multiples of 2, 3, 4, 5, 6 and 10.
These end-digit patterns become our divisibility rules.
Multiples of 10 always end in...
A handy set of shortcut tests:
| Divisor | Test | Example |
|---|---|---|
| 2 | Last digit 0, 2, 4, 6, 8 | 39,604 |
| 3 | Sum of digits divisible by 3 | 4,506 -> 4+5+0+6 = 15 |
| 4 | Last two digits divisible by 4 | 679,320 -> 20 |
| 5 | Last digit 0 or 5 | 50,005 |
| 6 | Divisible by both 2 and 3 | 5,922 |
| 10 | Last digit 0 | 50,000 |
Is 4,506 divisible by 3?
On a 1-100 grid, colour the multiples of 3 pink and the multiples of 4 green.
Multiples of 3 form diagonal stripes; multiples of 4 appear in every 4th cell. Numbers divisible by 6 are divisible by both 2 and 3, so they appear in both colourings.
Which of these is divisible by 6?
Each divisor pairs with its own test:
Pro tip: whether it's a two-digit number or an eight-digit number, the same rules of divisibility apply.
Which rule belongs to 5?
The LCM is the smallest number that is a multiple of two or more numbers.
Real-life example: Jupiter takes 12 years to orbit the Sun, Saturn takes 30 years. If they are aligned tonight, they line up again after the LCM of 12 and 30 years.
Math in real life: Astronomers use LCM to predict planetary alignments.
LCM stands for...
Method 1 - listing multiples.
Multiples of 12: 12, 24, 36, 48, 60, 72...
Multiples of 30: 30, 60, 90, 120...
LCM = 60.
Method 2 - prime factorisation.
12 = 2 x 2 x 3
30 = 2 x 3 x 5
LCM = 2 x 2 x 3 x 5 = 60 (common numbers written only once).
LCM(12, 30) = 60
What is the LCM of 12 and 30?
The HCF is the largest number that divides two or more numbers with no remainder.
Real-life example: Areeb has wooden planks 12 m and 30 m long. To cut them into equal pieces that are as long as possible, he uses the HCF.
Math Language: HCF - the largest number that divides each of the numbers exactly.
When do we use HCF instead of LCM?
Method 1 - listing factors.
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
HCF = 6.
Method 2 - prime factorisation.
12 = 2 x 2 x 3
30 = 2 x 3 x 5
HCF = 2 x 3 = 6 (common factors only).
Method 3 - Venn diagram. Multiply only the primes in the overlap.
HCF(12, 30) = 6
What is the HCF of 12 and 30?
Method 4 - repeated division.
Step 1: 12, 30 / 2 -> 6, 15.
Step 2: 6, 15 / 3 -> 2, 5. Now 2 and 5 share no common factor other than 1.
Step 3: multiply the common divisors: 2 x 3 = 6.
So Areeb cuts the planks into 6-metre pieces.
Areeb's planks are 12 m and 30 m. The longest equal piece is...
Quick checks:
Which is the prime factorisation of 44?
Match the sequences:
LCM and HCF practice:
Reflect: 1 is not prime (only 1 factor); 2 is the only even prime; primes have exactly 2 factors.
Let's reflect: a Venn diagram shows common factors in the overlap - that's where HCF lives.
Which sequence shows prime numbers?
Continue with quizzes, flashcards, or games when you are ready.