Finding THE Highest Common Factor (HCF) and THE Lowest Common Multiple (LCM)
What is the Highest Common Factor (HCF)?
The Highest Common Factor is the largest integer that two or more number can be divided by.
example: Find the lcm of 16, 24 and 36
Method 1: factorisation
Factors of 16: 1, 2, 4, 8, 16
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Therefore, HCF (16,24,36) = 4
METHOD 2: PRIME FACTORISATION
List all prime factors for each number.
Prime Factorisation of 16 is: 2 x 2 x 2 x 2
Prime Factorisation of 24 is: 2 x 2 x 2 x 3
Prime Factorisation of 36 is: 2 x 2 x 3 x 3
Highlight the prime factors that occur in all the prime factorisations of 16, 24 and 36.
The new list is 2 and 2.
Multiply these factors together to find the HCF.
HCF = 2 x 2 = 4
Therefore, HCF (16,24,36) = 4
METHOD 3: LADDER METHOD (fastest method for bigger numbers)
Write your numbers in a row.
Divide your numbers by prime numbers as long as all numbers are evenly divisible by that prime factor.
Ladder
What is the LEAST Common Multiple (lcm)?
The Least Common Multiple is the smallest positive integer that is divisible by two or more numbers. In other words, it is the first number that comes first in the times tables of two or more numbers.
example: Find the lcm of 16, 24 and 36
Method 1: Listing Multiples
Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.
Multiples of 16: 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176
Multiples of 24: 24, 48, 72, 96, 120, 144, 168, 192
Multiples of 36: 36, 72, 108, 144, 180, 216
Therefore, LCM(16, 24, 36) = 144
METHOD 2: PRIME FACTORISATION
List all prime factors for each number.
Prime Factorisation of 16 is: 2 x 2 x 2 x 2 => 2^4
Prime Factorisation of 24 is: 2 x 2 x 2 x 3 => 2^3 x 3^1
Prime Factorisation of 36 is: 2 x 2 x 3 x 3 => 2^2 x 3^2
For each prime factor, find where it occurs most often as a factor and write it that many times in a new list.
The new list is 2, 2, 2, 2, 3, 3
Multiply these factors together to find the LCM.
LCM = 2 x 2 x 2 x 2 x 3 x 3 = 144
In exponential form: LCM = 2^4 x 3^2 = 144
LCM = 144
Therefore, LCM(16, 24, 36) = 144
METHOD 3: LADDER METHOD (fastest method for bigger numbers)
Write your numbers in a row.
Divide your numbers by prime numbers as long as at least two numbers are evenly divisible by that prime.
Ladder
Bring down any numbers that are not evenly divisible by the current factor.
When there are no more primes that evenly divide into at least two numbers you are done.
The LCM is the product of the numbers in the L shape, left column and bottom row (highlighted in yellow). 1 is ignored.
LCM = 2 x 2 x 2 x 3 x 2 x 3 = 144
Therefore, LCM(16, 24, 36) = 144
