Types of Numbers

Composite, Square, Cubic, and Triangular

In this lesson:

• Classify whole numbers as prime, composite, or neither
• Identify square and cubic numbers — and see them geometrically
• Discover triangular numbers and use the formula

What You Will Learn This Lesson

By the end of this lesson, you can:

1. Classify whole numbers: prime, composite, or neither
2. Identify square numbers and calculate
3. Identify cubic numbers and calculate
4. Use for triangular numbers
5. Recognise overlaps — numbers in more than one category

Is 561 a Prime Number? Apply the Tests.

Apply the divisibility tests from Lesson 1:

Digit sum: 5 + 6 + 1 = 12 → divisible by 3 → 561 = 3 × 187

It has factors other than 1 and itself — not prime.

We need a system for classifying all numbers.

List the Factors: What Do You Find?

Factors of 12: 1, 2, 3, 4, 6, 12 → 6 factors

Factors of 7: 1, 7 → 2 factors

Factors of 1: 1 → 1 factor

Three numbers, three different outcomes.

What natural name fits each case?

Three Categories: Prime, Composite, and Neither

Prime: exactly 2 factors — 1 and itself. e.g. 2, 3, 5, 7, 11, …

Composite: more than 2 factors. e.g. 4, 6, 8, 9, 10, 12, …

Neither: exactly 1 factor — only 1 is in this category.

Every even number except 2 is composite.

Divisibility Tests as Classification Tools

561: digit sum 12, div by 3 → factor 3 → Composite

293: not even ✗ digit sum 14 ✗ ends in 3 ✗
293÷7≈41.8 ✗ 2−9+3=−4 ✗ → Prime

Tests narrow the search — no need to try every divisor.

Quick Check: Prime, Composite, or Neither?

Classify each number — prime, composite, or neither?

1. 9
2. 15
3. 25
4. 2

Write your answers before advancing.

From Counting Factors to Geometric Shape

So far: classify by factor count.

Now: classify by how dots arrange in space.

Can dots fill a perfect square grid?

An n×n grid holds n² dots — a square number.

Square Arrays: Building the Pattern

Each new square adds a new row and column — the nth square has n rows and n columns.

Square Numbers: Formula and Key Examples

Square numbers: for whole number

First 10 square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

A square number is one whose square root is a whole number.

Is 144 a square number? — yes.

Cubic Arrays: Extending to 3D

• 1³ = 1 (a single unit cube)
• 2³ = 8 (a 2×2×2 block of 8 small cubes)
• 3³ = 27 (a 3×3×3 block of 27 small cubes)

Square numbers: n×n (2 dimensions). Cubic numbers: n×n×n (3 dimensions).

Cubic Numbers: Formula and Key Examples

Cubic numbers: for whole number

First 6 cubic numbers: 1, 8, 27, 64, 125, 216

, , , , ,

Can a number be both square and cubic?

Overlap: Numbers That Are Both Square and Cubic

Is 64 a square number? ✓

Is 64 a cubic number? ✓

64 is both square and cubic. Also: 1 = =

64 = 8² = 4³ is the simplest non-trivial overlap.

Quick Check: Square and Cubic Numbers

1. Is 81 a square number? (Find .)
2. Is 125 a cubic number? (Find .)
3. Is 1 both square and cubic? Why?

Commit to all three answers before moving on.

From Arrays to Staircases: Triangular Numbers

What if each row grows by one dot?

• Row 1: 1 dot
• Rows 1–2: 3 dots
• Rows 1–3: 6 dots
• Rows 1–4: 10 dots

The shape is a staircase — n dots on the bottom row.

Visualising Triangular Numbers as Dot Diagrams

, , ,

What pattern do you see in the differences between consecutive terms?

The Difference Pattern Leads to the Formula

= 1, 3, 6, 10, 15

Differences: 2, 3, 4, 5 — increasing by 1 each time

Is there a shortcut to compute this sum?

Pairing Terms to Derive the Formula

Each pair , , … sums to .
There are such pairs:

Verify: ✓ ✓

Is 36 a Triangular Number?

Solve

Try consecutive integers: ✓

So : ✓

Yes — 36 is the 8th triangular number.

(And 36 = 6² — so 36 is both square and triangular!)

Is 30 a Triangular Number?

Solve

Test consecutive pairs:

• — too small
• — too large

No whole number satisfies .

30 is NOT a triangular number.

The test is systematic: try consecutive pairs until one overshoots.

Your Turn: Classify These Numbers

Classify each as prime, composite, square, cubic, or triangular.
A number may fit more than one category.

• 21
• 30
• 45
• 50

Key Takeaways: Classifying Numbers by Type

✓ Prime: exactly 2 factors (1 and itself) — most odd numbers are NOT prime

✓ Composite: more than 2 factors — use divisibility tests to find one

1 is neither prime nor composite

Odd ≠ prime: 9, 15, 25, 35 are all composite

Key Takeaways: Shaped Number Sequences

✓ Square: — 1, 4, 9, 16, 25, 36, …

✓ Cubic: — 1, 8, 27, 64, 125, …

✓ Triangular: — 1, 3, 6, 10, 15, 21, …

Next lesson: these types become building blocks of sequences.