6 — Factors and Prime Numbers

Factors and Prime Numbers

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Every number is made up of smaller building blocks called factors! Think of factors like puzzle pieces that fit perfectly together to create a bigger number. Some numbers have many factors (composite numbers), while others have only two factors (prime numbers).

Tip

To find factors quickly: Start with 1 and the number itself, then test if 2, 3, 4, 5... divide evenly into your number. Stop when you reach the middle!

Factors and Prime Numbers · 2:26

Worked Example

Problem

Find all factors of 24.

Solution

1, 2, 3, 4, 6, 8, 12, 24

Explanation

  1. 1Start with 1 and 24 (every number has these).
  2. 2Test if 2 divides evenly: 24 ÷ 2 = 12 ✓ (so 2 and 12 are factors).
  3. 3Test 3: 24 ÷ 3 = 8 ✓ (so 3 and 8 are factors).
  4. 4Test 4: 24 ÷ 4 = 6 ✓ (so 4 and 6 are factors).
  5. 5Test 5: 24 ÷ 5 = 4.8 ✗ (not a whole number).
  6. 6We already found 6, so we're done!

Worked Example

Problem

Find the prime factorization of 60 using a factor tree.

Solution

60 = 2² × 3 × 5

Explanation

  1. 1Start with 60 and break it into two factors: 60 = 6 × 10.
  2. 2Break down 6: 6 = 2 × 3 (both prime, so stop here).
  3. 3Break down 10: 10 = 2 × 5 (both prime, so stop here).
  4. 4Collect all prime factors: 2, 3, 2, 5.
  5. 5Write in order with exponents: 2² × 3 × 5.

Question

List all factors of 18.

Show Answer
1, 2, 3, 6, 9, 18
Show Solution
  1. 1Start with 1 and 18.
  2. 2Test 2: 18 ÷ 2 = 9 ✓ (so 2 and 9 are factors).
  3. 3Test 3: 18 ÷ 3 = 6 ✓ (so 3 and 6 are factors).
  4. 4Test 4: 18 ÷ 4 = 4.5 ✗.
  5. 5Test 5: 18 ÷ 5 = 3.6 ✗.
  6. 6We already found 6, so we have all factors: 1, 2, 3, 6, 9, 18.

Question

List all factors of 20.

Show Answer
1, 2, 4, 5, 10, 20
Show Solution
  1. 1Start with 1 and 20.
  2. 2Test 2: 20 ÷ 2 = 10 ✓ (so 2 and 10 are factors).
  3. 3Test 3: 20 ÷ 3 = 6.67 ✗.
  4. 4Test 4: 20 ÷ 4 = 5 ✓ (so 4 and 5 are factors).
  5. 5We already found 5, so we have all factors: 1, 2, 4, 5, 10, 20.

Question

Determine if each number is prime or composite: 29, 33, 51

Show Answer
29 is prime, 33 is composite, 51 is composite
Show Solution
  1. 1Check 29 - Test dividing by 2, 3, 4, 5... up to √29 ≈ 5.4. None divide evenly, so 29 has only factors 1 and 29 = prime.
  2. 2Check 33 - Test 3: 33 ÷ 3 = 11 ✓. Since 33 has factors 1, 3, 11, and 33, it's composite.
  3. 3Check 51 - Test 3: 51 ÷ 3 = 17 ✓. Since 51 has factors 1, 3, 17, and 51, it's composite.

Question

Find the prime factorization of 40.

Show Answer
40 = 2³ × 5
Show Solution
  1. 1Start with 40 and divide by the smallest prime (2): 40 ÷ 2 = 20.
  2. 2Continue dividing by 2: 20 ÷ 2 = 10.
  3. 3Continue dividing by 2: 10 ÷ 2 = 5.
  4. 45 is prime, so we stop.
  5. 5Count the 2's: we used 2 three times, so 40 = 2³ × 5.

Question

Find the prime factorization of 72.

Show Answer
72 = 2³ × 3²
Show Solution
  1. 1Divide by 2: 72 ÷ 2 = 36.
  2. 2Divide by 2: 36 ÷ 2 = 18.
  3. 3Divide by 2: 18 ÷ 2 = 9.
  4. 49 won't divide by 2, try 3: 9 ÷ 3 = 3.
  5. 5Divide by 3: 3 ÷ 3 = 1.
  6. 6Count factors: three 2's and two 3's, so 72 = 2³ × 3².

Question

List all factors of 45.

Show Answer
1, 3, 5, 9, 15, 45
Show Solution
  1. 1Start with 1 and 45.
  2. 2Test 2: 45 ÷ 2 = 22.5 ✗.
  3. 3Test 3: 45 ÷ 3 = 15 ✓ (so 3 and 15 are factors).
  4. 4Test 4: 45 ÷ 4 = 11.25 ✗.
  5. 5Test 5: 45 ÷ 5 = 9 ✓ (so 5 and 9 are factors).
  6. 6Test higher numbers, but we already found 9, so we have all factors: 1, 3, 5, 9, 15, 45.

Tip

Remember: Prime numbers have exactly 2 factors (1 and themselves), while composite numbers have more than 2 factors. Every composite number can be broken down into prime factors - like finding the DNA of numbers!