Prime or composite?
Recall: a prime number has exactly two positive factors, 1 and itself. A composite number has more than two positive factors. The number 1 is neither prime nor composite. Classify 1, 2, 9, 13 and 21 before continuing.
Learning goals
- Classify positive integers and distinguish prime factors from other factors.
- Express a composite number as a product of prime powers.
- State and interpret both existence and uniqueness in the Fundamental Theorem of Arithmetic.
- Use prime factorisation to explain divisibility, terminal-digit and compositeness results.
Concept lesson
Divisibility and factor trees
Use divisibility rules to begin a factor tree for 84. Continue splitting every composite branch until every terminal factor is prime.
Can every composite number be built from primes?
Think of prime numbers as the indivisible building blocks of positive integers. Multiplying primes creates composite numbers: 2 x 2 x 3 gives 12, while 3 x 5 x 7 gives 105. Try several composite numbers. Each can be broken down until only primes remain. The important question is not only whether this can always be done, but whether the final collection of primes is fixed.
Prime factors and complete prime factorisation
A prime factor of a number is a factor that is itself prime. Prime factorisation is complete only when every factor is prime. The number 1 may appear in an ordinary multiplication, but it is not prime and is not written in prime-power form. For example, 60 = 2 x 2 x 3 x 5 = 2^2 x 3 x 5.
From a factor tree to prime-power form
Different correct factor trees may begin differently, but they finish with the same prime factors. After reaching primes, arrange equal primes together and use exponents. Always multiply the prime powers once to verify the original number.
Fundamental Theorem of Arithmetic
Every composite number can be written as a product of primes, and its prime factorisation is unique apart from the order of the prime factors. The first part guarantees existence; the second guarantees uniqueness. Thus 2 x 3 x 5 and 5 x 2 x 3 represent the same prime factorisation of 30.
Why prime factors control terminal digits
A positive integer ends in 0 only if it is divisible by 10, so its prime factorisation must contain both 2 and 5. If a power is built only from the prime 2, no exponent can introduce a factor 5. Uniqueness of prime factorisation makes this impossibility conclusive.
Prime factorise 540
Express 540 as a product of prime powers.
Detect and repair an incomplete factorisation
A student writes 360 = 3 x 120 and stops. The product is correct, but 120 is composite, so this is not a prime factorisation.
Determine missing exponents in a prime-power form
If 360 = 2^x x 3^y x 5^z, find x, y and z.
Can a power of 12 end in zero?
Check whether 12^n can end in 0 for any natural number n.
Complete the prime factorisation of 756
Complete: 756 = 2 x 378 = 2 x 2 x 189. Continue factorising 189, then write the result in prime-power form. Check that every final factor is prime and verify by multiplication.
Do two factor trees disagree?
One factor tree for 90 begins with 9 x 10; another begins with 6 x 15. Complete both. Arrange the final prime factors in ascending order and explain why the theorem says the results agree.
Reason about a terminal digit
Can 18^n end in 5 for any natural number n? Use these prompts: Which prime factor must a number ending in 5 contain? What primes occur in 18^n? How does uniqueness justify the conclusion?
Explain why a constructed number is composite
Explain why 7 x 11 x 13 + 13 is composite without fully multiplying the expression. Look for a common factor and write the expression as a product of two integers greater than 1.
One-mark pattern: state the theorem
A complete one-mark answer must mention both facts: every composite number has a prime factorisation, and that factorisation is unique apart from order.
Two-mark pattern: prime powers and comparison
For an expression such as N = 2^x x 3^y x 5^z, first prime factorise N, write repeated primes with exponents, and then compare corresponding exponents. Show the factorisation rather than giving only x, y and z.
Must remember
1 is neither prime nor composite. A prime factorisation contains only primes. Equal primes may be grouped using exponents. Every composite number has one prime factorisation apart from order. Prime factors can be used to decide divisibility and possible terminal digits.
Exit ticket
(1) Express 1386 as a product of prime powers. (2) State the Fundamental Theorem of Arithmetic. (3) Decide whether 24^n can end in 0 for a natural number n and justify your answer using prime factors.