1.02 GCF and LCM
Greatest common factor (GCF)
Some numbers share the same factor. For example $2$2 is a factor of $10$10 and $24$24. We call these common factors.
When we are asked to find the greatest common factor (GCF) between two or more numbers, we are being asked what is the biggest number that they can both be divided by and that leaves no remainder (or leftovers).
Worked examples
Question 1
Evaluate: What is the greatest common factor between $12$12 and $24$24?
Think: The factors of $12$12 are $1,2,3,4,6$1,2,3,4,6 and $12$12.
The factors of $24$24 are $1,2,3,4,6,8,12$1,2,3,4,6,8,12 and $24$24.
So, $1,2,3,4,6$1,2,3,4,6 and $12$12 are all common factors but $12$12 is the greatest common factor because it is the biggest number.
Do: The GCF is $12$12.
Let's look at another example.
Question 2
Evaluate: What is the greatest common factor between $9$9 and $15$15?
Think: The factors of $9$9 are $1$1, $3$3 and $9$9. The factors of $15$15 are $1,3,5$1,3,5 and $15$15.
Do: The GCF is $3$3.
Least common multiple (LCM)
A multiple is the result of multiplying a number by an integer.
Remember, some numbers share multiples. For example, $12$12 is a common multiple of $2$2 and $3$3 because $2\times6=12$2×6=12 and $3\times4=12$3×4=12.
When we are asked to find the least common multiple, we are being asked to find the smallest multiple that is shared by two numbers.
Worked examples
Question 3
Evaluate: What is the least common multiple of $6$6 and $8$8?
Think: Multiples of $6$6 are: $6,12,18,24$6,12,18,24 and $30$30. Multiples of $8$8 are $8,16,24,32$8,16,24,32 and $40$40.
Since $24$24 is the first common number to appear between the two sets, $24$24 is the least common multiple.
Do: LCM = $24$24.
Ok, let's try another one.
Question 4
Evaluate: What is the least common multiple of $10$10 and $12$12?
Think: Multiples of $10$10 are $10,20,30,40,50$10,20,30,40,50 and $60$60. Multiples of $12$12 are $12,24,36,48$12,24,36,48 and $60$60.
Do: LCM = $60$60.
Practice questions
Question 5
Consider the numbers $36$36 and $4$4.
List all the factors of $36$36. Write them all on the one line separated by a comma.
List all the factors of $4$4. Write them all on the one line separated by a comma.
Hence find the greatest common factor of $36$36 and $4$4.
Question 6
What is the greatest common factor of $115$115 and $55$55?
Question 7
What is the least common multiple of $7$7 and $4$4?
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