Number Theory

Lesson

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 highest common factor (HCF) 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 left overs).

**Evaluate: **What is the highest 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 highest common factor because it is the biggest number.

**Do:** The HCF is $12$12.

Let's look at another example.

**Evaluate:** What is the highest 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 HCF is $3$3.

A multiple is the result of multiplying a number by an integer (for more info see Looking at Multiplication Sets).

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.

We we are asked to find the lowest common multiple, we are being asked to find the smallest multiple that is shared by two numbers.

**Evaluate:** What is the lowest 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 lowest common multiple.

**Do: **LCM = $24$24.

Ok, let's try another one.

**Evaluate:** What is the lowest 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.

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 highest common factor of $36$36 and $4$4.

What is the highest common factor of $115$115 and $55$55?

What is the lowest common multiple of $7$7 and $4$4?

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