Factors, Multiples, HCF and LCM
A factor of a number divides into it exactly, with no remainder, a multiple of a number is any number in its times table, and a prime number has exactly two factors, 1 and itself, with every whole number greater than 1 breaking down into a unique product of prime factors. The highest common factor (HCF) of two numbers is the largest number that is a factor of both, and the lowest common multiple (LCM) is the smallest number that is a multiple of both, used to simplify fractions and split quantities into identical groups.
Method
- To list the factors of a number, work through the whole numbers from 1 upward, checking which divide in exactly, and pair each factor with its 'partner' factor.
- To list the multiples of a number, multiply it by 1, 2, 3, 4 and so on in turn.
- To find the prime factors of a number, repeatedly divide by the smallest prime number that divides in exactly, until only 1 remains, then write the result in index form.
- To find the HCF of two numbers, write each as a product of prime factors and multiply together the lowest power of each prime that appears in both factorisations.
- To find the LCM of two numbers, write each as a product of prime factors and multiply together the highest power of every prime that appears in either factorisation.
- Use a factor tree or repeated division to organise prime factorisation clearly, and always check the final list of primes multiplies back to the original number.
- Recognise real-life HCF problems (splitting into equal groups with nothing left over) and LCM problems (events that repeat and need to coincide again).
Worked example
A youth club has 84 packets of crisps and 126 cans of drink. They want to make identical party bags, each containing the same number of packets of crisps and the same number of cans of drink, using all the crisps and drinks with none left over. Work out the greatest number of party bags that can be made, and how many packets of crisps and cans of drink would be in each bag.
- Write 84 as a product of prime factors: 84 = 2 x 2 x 3 x 7, which is 2^2 x 3 x 7.
- Write 126 as a product of prime factors: 126 = 2 x 3 x 3 x 7, which is 2 x 3^2 x 7.
- The greatest number of identical bags is the HCF of 84 and 126, found by multiplying the lowest power of each prime that appears in both factorisations: 2 (power 1), 3 (power 1) and 7 (power 1).
- HCF = 2 x 3 x 7 = 42, so the greatest number of identical party bags is 42.
- Divide each total by the HCF to find the contents of each bag: 84 / 42 = 2 packets of crisps, and 126 / 42 = 3 cans of drink per bag.
Practice questions
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Q1List all the factors of 56.Show answer
Answer: 1, 2, 4, 7, 8, 14, 28, 56
Q2Write down the first five multiples of 13.Show answer
Answer: 13, 26, 39, 52, 65
Q3Write 198 as a product of its prime factors, giving your answer in index form.Show answer
Answer: 2 x 3^2 x 11
Q4Find the HCF of 72 and 108.Show answer
Answer: 36
Q5Find the LCM of 18 and 24.Show answer
Answer: 72
Q6Josh waters his tomato plants every 4 days and his roses every 6 days. He waters both today. Work out how many days until he next waters both the tomatoes and the roses on the same day.Show answer
Answer: 12 days (the LCM of 4 and 6)
Q7A teacher has 90 red pens and 135 blue pens. She wants to put them into identical trays, each with the same number of red pens and the same number of blue pens, with none left over. Work out the greatest number of trays she could use.Show answer
Answer: 45 trays (the HCF of 90 and 135)
Q8Explain why 1 is not a prime number.Show answer
Answer: A prime number must have exactly two factors (1 and itself), but 1 only has one factor, itself, so it does not meet the definition of a prime number.
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
The numbers 60 and 84 are given. (a) Write each number as a product of its prime factors. (b) Use your answer to part (a) to find the HCF of 60 and 84. (c) Use your answer to part (a) to find the LCM of 60 and 84.
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Three fairground rides all depart together at 2:00pm. The carousel departs every 6 minutes, the Ferris wheel every 10 minutes, and the log flume every 15 minutes. Work out the next time after 2:00pm that all three rides will depart together again.
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Two positive whole numbers have a highest common factor (HCF) of 8 and a lowest common multiple (LCM) of 240. One of the numbers is 40. Work out the other number.
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Free printable worksheet
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This topic is chapter 6 of KS3 Maths Workbook 1, the whole course as one free printable PDF.
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