The Fundamental Theorem of Arithmetic
Every composite number can be written as a product of primes in exactly one way, apart from the order. This is the Fundamental Theorem of Arithmetic.
Prime and composite numbers
A factor of a number divides it exactly, with no remainder. A prime number has exactly two factors: 1 and itself. Examples are 2, 3, 5, 7, 11 and 13. A composite number has more than two factors, like 4, 6, 9 and 12. The number 1 is neither prime nor composite. The number 2 is the only even prime.
The theorem
Fundamental Theorem of Arithmetic: every composite number can be written as a product of primes. This factorisation is unique, apart from the order of the primes. Unique means there is only one way. For example, 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5. You may write the primes in any order. You always get the same primes, the same number of times.
How to find the prime factorisation (division method)
You can also use a **factor tree**: split the number into two factors, and keep splitting until every branch ends in a prime.
- 1Divide the number by the smallest prime that divides it exactly. Start with 2.
- 2Keep dividing by 2 while the quotient is even. Then try 3, then 5, 7, 11 and so on.
- 3Stop when the quotient becomes 1.
- 4★Multiply all the prime divisors. Use powers for repeated primes, e.g. 2 × 2 × 2 = 2³.
★ key step
Example 1: write 3150 as a product of primes
Find the prime factorisation of 3150.
- 3150 ÷ 2 = 1575
- 1575 ÷ 3 = 525
- 525 ÷ 3 = 175
- 175 ÷ 5 = 35
- 35 ÷ 5 = 7
- 7 ÷ 7 = 1, so we stop.
- Primes used: 2 once, 3 twice, 5 twice, 7 once.
- Check: 2 × 9 × 25 × 7 = 18 × 25 × 7 = 450 × 7 = 3150.
3150 = 2 × 3² × 5² × 7
Example 2: can 15ⁿ end with the digit 0?
n is a natural number. Check whether 15ⁿ can end with the digit 0.
- A number ends with 0 only if it is divisible by 10, that is, by both 2 and 5.
- So its prime factorisation must contain both 2 and 5.
- 15 = 3 × 5, so 15ⁿ = 3ⁿ × 5ⁿ.
- The only primes in 15ⁿ are 3 and 5. The prime 2 is missing.
- By the uniqueness part of the Fundamental Theorem of Arithmetic, no other prime can appear in 15ⁿ.
15ⁿ can never end with the digit 0, for any natural number n.
A sweet shop packing 360 laddoos can read every possible equal box size from 360 = 2³ × 3² × 5. Online payments are kept safe using huge numbers made by multiplying two large primes. Finding those primes again is extremely hard.
1 marks: “What is the exponent (power) of 2 in the prime factorisation of 144?” 144 = 2⁴ × 3², so the exponent of 2 is 4.
2 marks: “Check whether 12ⁿ can end with the digit 0 for any natural number n.” 12ⁿ = 2²ⁿ × 3ⁿ. To end in 0 it needs the prime 5. By uniqueness of prime factorisation, 5 never appears, so 12ⁿ cannot end in 0.
2 marks: “Explain why 5 × 7 × 11 + 7 is a composite number.” 5 × 7 × 11 + 7 = 7 × (5 × 11 + 1) = 7 × 56 = 392. It has factors other than 1 and itself (7 and 56), so it is composite.
One number, one prime recipe. Like a dish made from fixed ingredients, every number is built from its primes in only one way.