The Fundamental Theorem of Arithmetic

This is possibly the most important aspect of numbers:

Every positive integer (whole number), except for the number 1, is either a prime number or can be written as a product of prime factors.

Here are the numbers up to 10:

2 – prime

3 – prime

4 – 2²

5 – prime

6 – 2×3

7 – prime

8 – 2³

9 – 3²

10 – 2×5

See if you can continue this list up to 20 or 30.

Why is this so important? Well, it can help us to find square numbers:

36 = 2²×3²                   a square number

81 = 3⁴                         a square number

144 = 2⁴×3²                 a square number

400 = 2⁴×5²                 a square number

3969 = 3⁴×7²               a square number

4356 = 2²×3²×11²        a square number

Have you spotted the pattern? All the powers are multiples of two.

Can you find a similar pattern for cubic numbers?

What about quartic numbers?

Quintic numbers?

Note: writing numbers as shown above is called 'writing as a product of prime factors in index form'.

Further reading: https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic