What would you like to learn today?

Fundamental Theorem of Arithmetic Explorer

Fundamental Theorem of Arithmetic

Every composite number has a unique prime factorisation — explore why this matters

🔍 Hook: Can two different sets of primes multiply to give the same number?

Interactive Factor Tree

Enter any number to see its unique prime factorisation.

Try:

360 COMPOSITE

Prime factorisation: 360 = 2³ × 3² × 5

222335

Distinct prime factors: 2, 3, 5

Visual Factor Tree

The Uniqueness Principle

No matter how you split a number into factors, you always arrive at the same set of primes. Try different factorisation paths below:

Path A: 360 = 2 × 180
360 → 2 × 180 → 2 × 2 × 90 → 2 × 2 × 2 × 45 → 2³ × 3² × 5

Path B: 360 = 4 × 90
360 → 4 × 90 → 4 × 9 × 10 → 2² × 3² × 2 × 5 → 2³ × 3² × 5

✓ Same Result!

360 = 2³ × 3² × 5

Always unique

HCF & LCM via Prime Factorisation

Enter two or three numbers to compute HCF and LCM step-by-step using prime factorisations.

Number A

Number B

Number C (optional)

Examples:

Step 1: Prime Factorisations

Step 2: Compare Prime Powers

Prime	12	18	HCF (min)	LCM (max)
2	2²	2¹	2¹	2²
3	3¹	3²	3¹	3²

HCF = 6

2 × 3

Product of smallest powers of common prime factors

LCM = 36
22 × 32
Product of greatest powers of all prime factors

Key Relationship: HCF × LCM = a × b

✓ Verified!

HCF(12,18) × LCM(12,18) = 6 × 36 = 216

12 × 18 = 216

HCF × LCM = a × b ✓

Proof by Contradiction: √p is Irrational

Walk through the proof step-by-step. Choose a prime to prove its square root is irrational.

123456

1 Assume the contrary

Assume, to the contrary, that √2 is rational.

Then we can write √2 = a/b, where a and b are integers, b ≠ 0, and a, b are coprime (no common factors other than 1).

🧪 Exploration Sandbox

Investigate: Can 4ⁿ ever end in 0? Can 6ⁿ end in 0? Explore patterns in prime factorisations.

Does aⁿ end in 0?

A number ends in 0 only if it's divisible by both 2 and 5 — meaning its prime factorisation must contain both 2 and 5.

Base (a)

Max power (n)

Prime factorisation of 4: 22
Contains 2: Yes  |  Contains 5: No

✗ 4ⁿ CANNOT end in 0

Missing 5 as a prime factor — 10 can never divide 4ⁿ.

n	4ⁿ	Last digit	Ends in 0?
1	4	4	No
2	16	6	No
3	64	4	No
4	256	6	No
5	1024	4	No
6	4096	6	No
7	16384	4	No
8	65536	6	No
9	262144	4	No
10	1048576	6	No

Composite Number Checker

Verify that expressions like 7×11×13+13 are composite by finding their prime factors.

Custom number

LCM Meeting Problem Simulator

Sonia takes 18 min, Ravi takes 12 min to complete one round. When do they meet again? Generalise with any two times.

Person A (min)

Person B (min)