Prime & prime factorization

1Summary 重點摘要

Prime = divisible only by 1 & itself.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Spot primesA number with exactly two factors, 1 and itself, is a prime. For example, 7 has only the factors 1 and 7, so 7 is prime, while 6 has the factors 1, 2, 3 and 6, so it is not. The small primes are 2, 3, 5, 7, 11, 13, 17 and 19, and memorising this string makes factorization much faster later. Note especially that 2 is the only even prime: every other even number carries the factor 2 as an extra beyond 1 and itself, so none can be prime. A common error is treating 1 as prime, or overlooking 2; decide honestly by listing every factor and counting them.
2. Step 2: Spot compositesA number with factors beyond 1 and itself is a composite. For example, 12 has the factors 1, 2, 3, 4, 6 and 12, six in all, more than two, so 12 is composite and can be split into smaller numbers multiplied together, such as 12 = 3 × 4. Composites are exactly the numbers that can be broken into prime factors, while primes cannot be split. Why tell the two classes apart? Because prime factorization is performed only on composites, and a wrong start sends the whole question in the wrong direction. A common error is missing a factor, such as overlooking the 6 of 12 and misjudging the composite as prime; hunt factors in pairs, checking one by one from 1 upward.
3. Step 3: The special case of 1The number 1 has only one factor, itself, so it is neither prime nor composite. By definition a prime needs two factors and a composite needs three or more, and with only one factor, 1 fits neither class — a special case outside both. Nor does 1 ever appear in a prime factorization, because multiplying by 1 changes nothing and serves no purpose. Keeping this in mind saves trouble in true-or-false questions. A common error is filing 1 under the primes, or writing an extra factor as in 2 × 2 × 3 × 1; whenever a 1 appears in the answer, strike it out, since 1 never takes part in a prime factorization.
4. Step 4: Draw a factor treeDraw a factor tree: split a composite into two factors multiplied together, keep splitting, and stop when every branch ends in a prime; drawn out, the process looks like a tree. For example 12 = 4 × 3, then 4 splits into 2 × 2, while 3 is already prime, so 12 = 2 × 2 × 3. Since 2 and 3 are prime and cannot be split further, the factorization is complete. Why must the splitting reach primes? Because a prime factorization requires every factor to be prime, and the job is unfinished while a composite remains. A common error is stopping halfway and presenting 12 = 4 × 3 as the final answer; before finishing, check every factor and confirm each is prime.
5. Step 5: Any path, same resultThe factor tree allows more than one route: 12 can also start as 6 × 2, with 6 then splitting into 2 × 3, and the result is again 2 × 2 × 3. No matter which factor pair starts the splitting, the prime factors come out the same, with only their order possibly different. This shows that the result of prime factorization is unique — the very reason it matters, and a deep rule of arithmetic to be studied further in secondary school; at primary level, trying several routes verifies it directly. A common error is fearing a wrong first choice, as if different routes gave different results; start from any pair with confidence, since the destination is fixed.
6. Step 6: Exponent formRepeated prime factors can be written more briefly with exponents: 12 = 2 × 2 × 3 = 2² × 3, and 36 = 2 × 2 × 3 × 3 = 2² × 3². The exponent counts how many times a factor is multiplied, and 2² reads as two to the second power, meaning two 2's multiplied. Why use exponents? Because long chains of repeated factors are clumsy, while exponent form is tidy, and it also eases comparison: at a glance, 36 holds two 2's and two 3's. A common error is treating the exponent as a multiplier, computing 2³ as 2 × 3 = 6 instead of 2 × 2 × 2 = 8; an exponent counts repetitions, so be clear about that before computing.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Primes have exactly two factors: 1 and themselves
• 1 is neither prime nor composite
• 2 is the only even prime
• Split the factor tree until every branch is prime
• The prime factorization is unique (write as 2² × 3)