Prime factors, common factors and common multiples
Week 5 · Prime factors, greatest common factor and least common multiple · 3 lessons
1
Factor trees and the greatest common factor
Discover Mathematics P5: Term 1, Unit 2, Lessons 6–10
GoalSplit a number into prime factors with a factor tree, and find the greatest common factor of two numbers.
New words
prime number · a number with exactly two factors, 1 and itself, such as 7composite number · a number with more than two factors, such as 12prime factorization · a number written as a product of prime numbersgreatest common factor (GCF) · the biggest number that is a factor of both numbers
Explanation
A prime number has exactly two factors: 1 and itself. The first primes are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. A composite number has more than two factors. The number 1 is neither prime nor composite. A factor tree splits a composite number into a factor pair, then splits every composite factor again, until only primes are left at the ends of the branches. For 60: 60 = 6 × 10, then 6 = 2 × 3 and 10 = 2 × 5, so 60 = 2 × 2 × 3 × 5. You can start with any pair and you still reach the same primes. To find the GCF of two numbers, list the factors of both and pick the greatest number in both lists, or multiply the primes the two factorizations share. For 24 = 2 × 2 × 2 × 3 and 36 = 2 × 2 × 3 × 3 the shared primes are 2, 2 and 3, so the GCF is 2 × 2 × 3 = 12.
Worked examples
Make a factor tree for 36. 36 = 4 × 9. Then 4 = 2 × 2 and 9 = 3 × 3. The leaves are all prime, so 36 = 2 × 2 × 3 × 3. Check: 2 × 2 × 3 × 3 = 4 × 9 = 36.
Start 90 with a different pair: 90 = 2 × 45. The 2 is prime, so it stops. 45 = 5 × 9 and 9 = 3 × 3. The primes are 2, 5, 3 and 3, so 90 = 2 × 3 × 3 × 5. Splitting 90 as 9 × 10 gives the same primes.
Find the GCF of 24 and 36. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. The common factors are 1, 2, 3, 4, 6 and 12, so the GCF is 12. A 24 cm by 36 cm card can be cut into squares of side 12 cm with nothing left over: 2 × 3 = 6 squares.
Class activity
Prime hunt. Write the numbers from 2 to 50 in rows on squared paper. Circle 2 and cross out all its other multiples. Then circle the next number that is not crossed out (3) and cross out its multiples. Keep going. The circled numbers are the primes up to 50. Count them.
Practice
1
Write all the prime numbers between 10 and 30.
11, 13, 17, 19, 23, 29.
2
Use a factor tree to write 48 as a product of primes.
