Identify highest common factors and least common multiples.
Number Framework Stage 8
Highest Common Factors (HCFs) are needed for a range of problems involving fractions and eventually algebra. They require good understanding of factors and the skill of instant recall of the basic multiplication facts (tables). Failure to know tables effectively shuts students out of working with factors and therefore highest common factor work.
Using Number Properties
Problem: “Brian, Peter, and Murray buy packets of cakes. Brian has 12 cakes, Peter15 cakes, and Murray 9 cakes. How many cakes are there in each packet? (Assume there is more than 1 in a packet.)”
Discuss why 12, 15, and 9 are all multiples of the number of cakes in a packet. So the number in a packet must be 3.
Problem: “This time Brian, Peter, and Murray buy packets of lollies. Brian has 60 lollies, Peter 40 lollies, and Murray 48 lollies. Write down the factors of 60, 40, and 48 and find the common factors.”
(Answer: 60 has factors 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. 40 has factors 1, 2, 4, 5, 8, 10, 20, 40. 48 has factors 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. The common factors in all lists are 2 and 4 – ignore 1.)
What is the biggest number of lollies that could be in each packet? (Answer: 4.)
Examples: Find the highest common factor for these sets of numbers. Draw factor trees only if you have to: 54, 36, 24
36, 72, 24
50, 80, 45
36, 42, 6 000
500, 15, 25 21, 54, 18 ...
Problem: “Maureen, Chen, and Anna open packets of lollies to put in bowls for a party. Maureen’s bowl has 68 lollies, Chen’s bowl has 51 lollies, and Anna’s bowl has 34. How many lollies are in each packet?”
(Possible answer: Using prime factors 68 = 2 x 34 = 2 x 2 x 17, 51 = 3 x 17, 34 = 2 x 17. So the HCF is 17. So one packet has 17 sweets.)
Problem: “Norrie wants to find the highest common factor of 372 and 558. He decides to write each number as a product of prime numbers. Find the HCF of 372 and 558.”
(Answer: By using divisibility tests 372 = 2 x 186 = 2 x 2 x 93 = 2 x 2 x 3 x 31. And 558 = 2 x 279 = 2 x 3 x 93 = 2 x 3 x 3 x 31. 2 and 31 are common factors so is 2 x 31 = 62. So 62 is the HCF.)
Examples: Find the HCFs of these sets of numbers: 195 and 312
312 and 408
162 and 522
208,130, and 234
378, 224, and 140 ...