Week 1 · Long division, prime factors, and the GCF and LCM with Venn diagrams · 3 lessons
2
Breaking a number into prime factors
Discover Mathematics P6: Term 1, Unit 1, Lessons 1–2
GoalWrite a number as a product of prime numbers with a factor tree.
New words
prime number · a number with exactly two factors, 1 and itselfcomposite number · a number with more than two factorsfactor tree · a tree that splits a number into smaller factorsprime factorization · a number written as a product of primes only
Explanation
A prime number has exactly two factors: 1 and itself. 2, 3, 5, 7, 11 and 13 are prime. The number 1 is not prime because it has only one factor. A number with more than two factors, like 12, is composite. To break a composite number into primes, build a factor tree: split the number into any two factors, then split those factors again, and stop when every branch ends in a prime. The primes at the ends of the branches, multiplied together, give the number. You can start with any pair of factors, and you always get the same primes. For example, 36 = 4 × 9 = 2 × 2 × 3 × 3.
★Remember
A prime has only the factors 1 and itself. Split until every branch ends in a prime. 1 is not prime.
Worked examples
Factor tree of 36. Split 36 into 4 × 9. Then 4 = 2 × 2 and 9 = 3 × 3. Every branch ends in a prime, so 36 = 2 × 2 × 3 × 3.
Divide by a prime
Number left
90
2
45
3
15
3
5
5
1
Repeated division (the ladder) gives the same result. Divide 90 by the smallest prime that fits, again and again, until you reach 1: 90 ÷ 2 = 45, 45 ÷ 3 = 15, 15 ÷ 3 = 5, 5 ÷ 5 = 1. So 90 = 2 × 3 × 3 × 5.
Start differently, get the same primes. Mina splits 84 into 7 × 12, then 12 into 3 × 4, then 4 into 2 × 2. Youssef starts with 4 × 21. Both finish with 84 = 2 × 2 × 3 × 7. Check: 4 × 3 × 7 = 84.
Class activity
Prime hunt. Write the numbers 2 to 50 on a sheet. Cross out 4, 6, 8 and every other multiple of 2 (but keep 2). Then cross out the multiples of 3, then of 5, then of 7. The numbers that stay are the primes. How many primes are there below 50?
Practice
1
List all the prime numbers less than 20.
2, 3, 5, 7, 11, 13, 17 and 19.
2
Draw a factor tree for 48 and write it as a product of primes.
