Venn diagrams of prime factors: finding the GCF and the LCM
Discover Mathematics P6: Term 1, Unit 1, Lessons 1–2
GoalUse a Venn diagram of prime factors to find the GCF and the LCM of two numbers.
New words
GCF · the greatest common factor: the biggest number that divides bothLCM · the least common multiple: the smallest number that both divide intoVenn diagram · two circles that overlap to show what is sharedrelatively prime · two numbers whose only common factor is 1
Explanation
Write the prime factors of two numbers in a Venn diagram. The primes that both numbers have go in the overlap; the primes that only one number has go in its own circle. Take 12 = 2 × 2 × 3 and 18 = 2 × 3 × 3. Both have a 2 and a 3, so those go in the middle. 12 has one extra 2, and 18 has one extra 3. The GCF is the product of the overlap: 2 × 3 = 6. The LCM multiplies every number written in the diagram, but a shared number is used only once: 2 × 3 × 2 × 3 = 36. If the overlap is empty, the numbers are relatively prime and their GCF is 1.
ƒFormula
GCF = product of the overlap
LCM = multiply every number in the diagram (a shared number once)
Worked examples
12 and 18. The overlap holds 2 and 3, so GCF = 2 × 3 = 6. All the numbers are 2, 3, 2 and 3, so LCM = 2 × 3 × 2 × 3 = 36. Check: 36 ÷ 12 = 3 and 36 ÷ 18 = 2.
20 = 2 × 2 × 5 and 30 = 2 × 3 × 5. They share 2 and 5, so GCF = 2 × 5 = 10. The diagram holds 2, 5, 2 and 3, so LCM = 2 × 5 × 2 × 3 = 60.
8 = 2 × 2 × 2 and 15 = 3 × 5 share no prime, so the overlap is empty. 8 and 15 are relatively prime, GCF = 1 and LCM = 8 × 15 = 120.
Class activity
Two-circle game. With a grown-up, pick two numbers below 40, such as 24 and 36. Each of you draws the Venn diagram of their primes on paper (use two hoops or two loops of string if you have them) and finds the GCF and the LCM. Compare answers and check that both numbers divide the LCM.
Practice
1
Use the primes of 24 = 2 × 2 × 2 × 3 and 36 = 2 × 2 × 3 × 3 to find the GCF and the LCM.
Shared: 2, 2, 3, so GCF = 12. Only 24: one 2. Only 36: one 3. LCM = 2 × 2 × 3 × 2 × 3 = 72.
