HCF and LCMPractise finding the highest common factor (HCF), sometimes called the greatest common divisor, and the lowest common multiple (LCM) of two numbers. 
This is level 7: Mixed application questions. You can earn a trophy if you get at least 9 correct and you do this activity online.
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Scan the QR code below to visit the online version of this activity.
https://www.Transum.org/go/?Num=337
Sieve Use the Sieve of Eratosthenes to find prime numbers.
Factor Trees An interactive and very visual way to break down a number into its prime factors.
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Level 1  Finding the highest common factor (HCF) of two numbers.
Level 2  Finding the lowest common multiple (LCM) of two numbers
Level 3  Finding the highest common factor (HCF) of large numbers.
Level 4  Finding the lowest common multiple (LCM) of large numbers
Level 5  Finding the HCF and LCM of three numbers
Level 6  Given the HCF and LCM find the numbers
Level 7  Mixed application questions
HCF and LCM given An Advanced Lesson Starter.
Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.
See the National Curriculum page for links to related online activities and resources.
The highest common factor (HCF) of two numbers is the largest number that divides exactly into both of the numbers.
You can Find the HCF of numbers by listing the prime factors of both numbers then multiplying together the factors that appear in both lists.
For example find the HCF of 24 and 36
24 = 2x2x2x3 and 36 = 2x2x3x3
so the HCF of 24 and 36 is 2x2x3 = 12
The lowest common multiple (LCM), or least common multiple, is the smallest number that both numbers divide into exactly.
You can Find the LCM of numbers by listing the prime factors of both numbers and then multiply all the prime factors of the larger number by those prime factors of the smaller number that are not already included.
For example find the LCM of 24 and 36
24 = 2x2x2x3 and 36 = 2x2x3x3
so the LCM of 24 and 36 is 2x2x3x3 x 2 = 72
A Venn diagram may help you with the task of finding the HCF and LCM of 24 and 36.
Express each number as the primes which multiplied together would give you that number. Write them in Venn diagram sets:
Show the sets intersecting
Multiply the numbers in the intersection of the sets to find the HCF, 2x2x3 = 12.
Multiply all the numbers in the overlapping sets diagram to find the LCM, 2x2x2x3x3 = 72.
Click here to see an animated demonstration of this cool way to find both the HCF and LCM of two numbers.
Advanced calculators have built in functions for finding the HCF and LCM of two numbers but there is a trick for finding the HCF using a modern scientific calculator.
If the two numbers are entered using the fraction template the calculator will express that fraction in its lowest terms. It does this by dividing numerator and denominator by their HCF.
For example to find the HCF of 24 and 36 enter 24/36 then press enter.
Considering the denominators, we now need to find what 24 was divided by to give 2. So dividing 24 by 2 gives 12 which is the HCF.
When you have found the HCF of the numbers a and b the LCM can be found using the following formula:
It is worth knowing that HCF is also known as GCD. If you are using a spreadsheet such as Excel there are functions named LCM and GCD for calculating the LCM and HCF.
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Transum,
Monday, May 13, 2024
"In my student days I worked my summers in America; at Camp Tamarack in New Jersey. I remember in 1979 being surrounded by large colourful insects called cicadas which can only be seen every 17 years.
In 2024, Illinois is experiencing a rare natural spectacle: the simultaneous emergence of cicada Broods XIII and XIX. This event, first recorded in 1803, won’t occur again until 2245. These periodical cicadas, which emerge either every 13 or 17 years, spend the majority of their lives underground, only surfacing to reproduce at the end of their long developmental cycles.
After mating, females lay approximately 500 to 600 eggs in woody plants before the adults die, their life above ground lasting merely a month. The eggs hatch six weeks later, and the nymphs burrow into the soil to begin their 13 or 17year cycle.
This synchronisation of 13 and 17year cycles is a perfect natural example of common multiples.
"