The Product Rules Of HCF And LCM

Understand and apply the relationship HCF × LCM = a × b for two integers.

8 lessons

c2a86edc...

Concept

The Golden Relationship

Introduction to the product rule.

When working with two positive integers ( $a$ and $b$ ), there is a powerful shortcut connecting their Highest Common Factor (HCF) and Least Common Multiple (LCM).

Instead of calculating both from scratch using tedious prime factorisation, you can use their mathematical relationship to quickly find one if you already know the other.

1 / 2

df772e02...

html

Product Formula

Formal statement of the two-number rule.

The Product Formula

HCF (a, b) \times LCM (a, b) = a \times b

Condition: Valid ONLY for exactly TWO positive integers.

cc040670...

content

Product Formula

Formal proof of the two-number rule.

For any two positive integers $a$ and $b$ :

\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b

This means the product of two numbers is equal to the product of their HCF and LCM.

1 / 6

3f56f25e...

Worked Example

Calculating LCM via HCF

Example 3: 96 and 404.

Problem

Find the HCF of $96$ and $404$ by the prime factorisation method. Hence, find their LCM.

1 / 5

fda2edf3...

content

Warning: Three Numbers

Cautionary notes

Caution: This Rule Works Only for Two Numbers

For two positive integers, we have:

\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b

But for three numbers, this is not generally true:

\text{HCF}(p,q,r) \times \text{LCM}(p,q,r) \neq p \times q \times r

1 / 3

02b0ca37...

reading

Using the Given HCF

Exercise 1.1 Q4: Given HCF(306, 657) = 9, find LCM.

0 of 3 blanks filled

We know that for any two positive integers $a$ and $b$ :

$\text{HCF}(a,b) \times \text{LCM}(a,b)=a \times b$

So,

$\text{LCM}(a,b)=\frac{a \times b}{\text{HCF}(a,b)}$

Here,

$a=$ , $b=657$ , and $\text{HCF}(306,657)=$ .

Now substitute:

$\text{LCM}(306,657)=\frac{306 \times 657}{9}$

Therefore, the final calculated LCM is .

4ed4a989...

quiz

Three Number Trap

Test if student falsely applies the rule to 3 numbers.

1 / 5

If $a$ , $b$ , and $c$ are three positive integers, which of the following statements is ALWAYS true according to the text?

← previousFinding HCF And LCM Using Prime Factoriz...next →Revisiting Irrational Numbers: Proving P...

Concept

The Golden Relationship

Introduction to the product rule.

When working with two positive integers ( $a$ and $b$ ), there is a powerful shortcut connecting their Highest Common Factor (HCF) and Least Common Multiple (LCM).

Instead of calculating both from scratch using tedious prime factorisation, you can use their mathematical relationship to quickly find one if you already know the other.

1 / 2

content

Product Formula

Formal proof of the two-number rule.

For any two positive integers $a$ and $b$ :

\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b

This means the product of two numbers is equal to the product of their HCF and LCM.

1 / 6

Worked Example

Calculating LCM via HCF

Example 3: 96 and 404.

Problem

Find the HCF of $96$ and $404$ by the prime factorisation method. Hence, find their LCM.

1 / 5

content

Warning: Three Numbers

Cautionary notes

Caution: This Rule Works Only for Two Numbers

For two positive integers, we have:

\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b

But for three numbers, this is not generally true:

\text{HCF}(p,q,r) \times \text{LCM}(p,q,r) \neq p \times q \times r

1 / 3

reading

Using the Given HCF

Exercise 1.1 Q4: Given HCF(306, 657) = 9, find LCM.

0 of 3 blanks filled

We know that for any two positive integers $a$ and $b$ :

$\text{HCF}(a,b) \times \text{LCM}(a,b)=a \times b$

So,

$\text{LCM}(a,b)=\frac{a \times b}{\text{HCF}(a,b)}$

Here,

$a=$ , $b=657$ , and $\text{HCF}(306,657)=$ .

Now substitute:

$\text{LCM}(306,657)=\frac{306 \times 657}{9}$

Therefore, the final calculated LCM is .

quiz

Three Number Trap

Test if student falsely applies the rule to 3 numbers.

1 / 5

If $a$ , $b$ , and $c$ are three positive integers, which of the following statements is ALWAYS true according to the text?