Prime Numbers: The Building Blocks of Maths
Definition: A Prime Number is a whole number greater than 1 that has exactly two factors: 1 and itself.
1. Identifying Prime Numbers
To check if a number is prime, ask yourself: "Can I divide this number by anything other than 1 and itself without leaving a remainder?"
7 is Prime
Factors: 1, 7.
Reason: Nothing else goes into 7.
9 is NOT Prime
Factors: 1, 3, 9.
Reason: It can be divided by 3.
13 is Prime
Factors: 1, 13.
Reason: No even numbers, 3s, or 5s go into it.
15 is NOT Prime
Factors: 1, 3, 5, 15.
Reason: It is divisible by 3 and 5.
⚠️ The Number 1 is NOT Prime!
By definition, a prime must have exactly two factors. The number 1 only has one factor (itself), so it is not prime.
2. Primes up to 30
| Number |
Is it Prime? |
Why? |
| 2 | ✅ Yes | The only even prime number. |
| 3 | ✅ Yes | Factors are 1 and 3. |
| 5 | ✅ Yes | Factors are 1 and 5. |
| 21 | ❌ No | \(3 \times 7 = 21\) |
| 23 | ✅ Yes | No smaller numbers divide into it. |
| 27 | ❌ No | \(3 \times 9 = 27\) |
| 29 | ✅ Yes | Factors are 1 and 29. |
3. The "Fundamental Theorem of Arithmetic"
This sounds fancy, but it just means that every number that isn't prime can be broken down into a product of prime numbers. We call this Prime Factorization.
Example: 60
\(60 = 2 \times 30\)
\(60 = 2 \times 2 \times 15\)
\(60 = 2 \times 2 \times 3 \times 5\)
In index form: \(2^2 \times 3 \times 5\)
4. How to Find Large Primes
To test if a number like 101 is prime, you only need to check if it's divisible by primes up to its square root (\(\sqrt{101} \approx 10\)).
- Divisible by 2? No (it's odd)
- Divisible by 3? No (\(1+0+1=2\), not a multiple of 3)
- Divisible by 5? No (doesn't end in 0 or 5)
- Divisible by 7? No (\(101 \div 7 = 14.4...\))
Since none of these work, 101 is Prime!