Now the product property becomes a problem-solving tool.

The equation HCF $\times$ LCM = a $\times$ b has four quantities.

If you know any three, you can find the fourth by:

Rearranging the formula
Dividing to isolate the unknown

This section walks through both standard problem types:

Goal	Given Information
Find the other number	HCF, LCM, and one number
Find the LCM	HCF and both numbers

1. Setting Up the Equation

Setting Up the Equation 📝

The first step in these problems is recognising which three quantities you know and setting up the equation correctly.

Identifying the given values ( $H C F$ , $L C M$ , or the numbers $a$ and $b$ ).
Placing them correctly into the formula: $HCF(a, b) \times LCM(a, b) = a \times b$ .

This sounds simple, but errors in setup lead to wrong answers.

Let's see if you can handle this:

📋 Given Info

Problem:

The HCF of two numbers is 23, their LCM is 1449, and one of the numbers is 161.

Find the other number.

Given Information:

$H C F = 23$
$L C M = 1449$
One number ( $a$ ) $= 161$
Other number ( $b$ ) $= ?$ ?

✍️ Question

Your Task ✍️

Write the product property equation for this problem and substitute the known values.

What equation do you need to solve?

(Do NOT solve it yet — just set it up.)

Setting Up the Problem

To find an unknown number or value using the relationship between HCF and LCM, start by stating the formal property clearly.

Step 1: Write the Product Property

The fundamental rule connecting these values is:

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

(HCF times LCM equals the product of the two numbers)

Note: The product of the Highest Common Factor and the Lowest Common Multiple of two numbers is always equal to the product of the two numbers themselves.

✍️ MCQ

Choose one

In the formula

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

, what do

a

and

b

represent?

The Four QuantitiesHCF, LCM, and the two original numbers

This equation connects four specific quantities:

HCFOne of the four quantities you work with of the numbers
LCMOne of the four quantities you work with of the numbers
The first number ( $a$ )
The second number ( $b$ )

Key Insight: If you know any threeThree known values is enough to solve of these quantities, you can use algebraic rearrangement to find the missing fourth onegoalUse basic algebra to find the fourth!

✍️ Yes/No

Yes or No?

If we know the HCF, the LCM, and one of the numbers (

a

), can we find the other number (

b

)?

Step 2: Identify the Knowns and Unknowns

To solve this systematically, start by organizing the information provided in the problem. This helps ensure you have all the necessary components for the formula.

Given Values:

HCF = 23List this value first
LCM = 1449Write down your L C M
First Number ( $a$ ) = 161The number given in the problem

To Find:

The Second Number ( $b$ ) = ?

Identifying exactly what we know and what we are looking for is the first step to staying organized during a calculation.Prevents putting wrong values into the formula

✍️ Yes/No

Yes or No?

Do we have enough information to find the value of

b

?

Step 3: Substitute the ValuesThe key technique in this step

With our three known values identified, we can now plug them directly into the Product Formula.

The Formula: $HCF \times LCM = a \times b$

Substituting the numbers:

23 \times 1449 = 161 \times b

(Turn the formula into a solvable equation)

By replacing the terms HCF, LCM, and the known variable $a$ with their numerical values, we create an equation where $b$ is our only unknown.

✍️ MCQ

Choose one

To isolate

b

, we should:

Step 4: Rearrange to Isolate the UnknownSaves a lot of time in exams

To find the value of $b$ , we need to isolate it on one side of the equation. We do this by moving the known number (161) to the other side using division.

b = \frac{23 \times 1449}{161}

(H C F times L C M in the numerator)

💡 Key Insight: The Rearrangement Pattern

When using the product formula to find a missing number, you will always follow this pattern:

$Unknown Number = \frac{HCF \times LCM}{Known Number}$

Numerator: The product of the HCF and LCM.Always in the numerator
Denominator: The known number given in the problem.The known number goes here

Using the Product Formula

When you know the HCF and LCM of two numbers ( $a$ and $b$ ), but one of the numbers is missing, you can use a handy shortcut formula.

General formula:

To find the unknown number, use:

\text{other number} = \frac{\text{HCF} \times \text{LCM}}{\text{known number}}

Or equivalently, if $a$ is the known number and $b$ is the unknown:

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

(This formula connects HCF, LCM, and both numbers)

Memory Tip: The product of the HCF and LCM always stays on topMultiply HCF and LCM for the numerator (the numerator), and the number you already have goes on the bottomThe known number becomes the divisor (the denominator).

✍️ MCQ

Choose one

To find the missing number

b

, which value must be in the denominator?

A Flexible Tool

This formula is totally flexibleWorks in any direction to find missing values! As long as you have three out of the four valuesAny three values let you solve for the fourth ( $a, b, HCF, LCM$ ), you can always find the fourth one by "moving the pieces around."

Rearranging to find LCM or HCF:

To find the LCM: If you know both numbers $a$ and $b$ and their HCF:
$\text{LCM} = \frac{a \times b}{\text{HCF}}$
(Divide the product of both numbers by their HCF)
To find the HCF: If you know both numbers $a$ and $b$ and their LCM:
$\text{HCF} = \frac{a \times b}{\text{LCM}}$
(Same formula, just solving for a different piece)

✍️ T/F

True or False?

True or False: To find the HCF of two numbers

a

and

b

, we can use the formula:

\text{HCF} = \frac{a \times b}{\text{LCM}}

2. Solving for the Missing Number

Let's complete the computation!

From our previous work, we set up the equation and found:

$b = \frac{23 \times 1449}{161}$

The main challenge here is the multi-digit arithmetic.
After finding the answer, you'll need to verify that it actually works in the product formula.

✍️ Question

Your Task ✏️

Compute the value of $b$ using our formula:

b = \frac{23 \times 1449}{161}

solve this

Show your working. Then verify your answer:

Is $\text{HCF}(161, b) = 23$ ?
Is $\text{LCM}(161, b) = 1449$ ?

Step-by-Step Multiplication: $23 \times 1449$

To make this calculation easier, we can break downSplit 1449 into simpler pieces to work with the larger number into manageable parts:

1. Break $1449$ into $1400 + 49$

2. Calculate the first part:

$23 \times 1400 = (23 \times 14) \times 100$
Since $23 \times 14 = 322$ , then $322 \times 100 = 32, 200$

Breaking down numbers is a great strategy to manage large products without a calculator.

✍️ MCQ

Choose one

If

23 \times 50 = 1150

, then

23 \times 49 = 1150 - \dots

Step 2: Multiplication Shortcut

To calculate the second part, $23 \times 49$ , we can use a subtraction shortcutUse 50 minus 1 instead of 49 directly by relating it to $23 \times 50$ :

$23 \times 49 = 23 \times (50 - 1)$
$23 \times 49 = (23 \times 50) - 23$
$23 \times 49 = 1150 - 23 = 1, 127$

Now we have our two intermediate pieces: $32, 200$ and $1, 127$ .

✍️ FIB

Fill in the blank

Calculate

32,200 + 1,127

.

Final Result

By adding the two parts together, we find the total product:

$23 \times 1449 = 32, 200 + 1, 127 = 33, 327$

Key Takeaway: In this context, this value represents the product of the HCF and LCMThis result equals HCF times LCM. Breaking down the multiplication made finding this large number much more manageable!Exams often ask you to find HCF or LCM using this property

Final Step: Finding the Unknown Number $b$

We are at the final stage of solving for our unknown number!

The Calculation Path

From our previous steps, we have the product of HCF and LCMCalculate HCF times LCM before anything else as $33, 327$ . To isolate the unknown number ( $b$ ), we use the relationship:

$Unknown Number (b) = \frac{HCF \times LCM}{Known Number (a)}$

Computing $33, 327 \div 161$ : We will break this large division down step-by-stepAvoid calculation errors with large numbers to ensure accuracy.

✍️ FIB

Fill in the blank

Calculate:

161 \times 200

Division by Estimation

To make the large division easier, let's look at how close we can get using a round number like $200$ :

Estimate: $161 \times 200 = 32,200$ Round numbers make estimation easier
Difference: $33,327 - 32,200 = 1,127$ remainderFind the remainder to complete the calculation

This shows that $32, 200$ is very close to our target. Our final answer for $b$ will be $200$ plus whatever $1, 127 \div 161$ equals.

✍️ MCQ

Choose one

If

1 \times x

ends in

7

, what is

x

?

Completing the Division

By checking the unit digits ( $1 \times 7 = 7$ ), we can determine the remaining part of the quotient:

$161 \times 7 = 1, 127$
Since there is no remainder leftNo remainder means the quotient is exact, we combine our parts:

$33, 327 \div 161 = 200 + 7 = 207$

The missing number $b = 207$ .

💡 Golden RuleA key principle for checking your work: The Whole Number Check

Since HCF and LCM apply to integers, your result for a missing number must always be a whole numberFactors and multiples only work with integers. If you find yourself with a decimal or a remainder, stop and re-check your arithmeticA decimal result means there's a calculation error!

✍️ Yes/No

Yes or No?

Is

45.2

a valid answer for the 'other number' in this type of problem?

Verification

Now that we've found our second number is 207, let's double-check everything. It's always a great idea to verify your answerAlways check your work to be certain to be 100% sure.

First, let's break down (factorise) our first known number, 161givenSee what prime factors we're working with:

$161 \div 7 = 23$
So, the factorisation is: $161 = 7 \times 23$

✍️ Yes/No

Yes or No?

Is

207

divisible by

3

?

Factorising 207

To factorise $207$ , we find its prime parts:

$207 \div 3 = 69$
$69 \div 3 = 23$ Keep dividing until you hit a prime (23 is a prime number)

So, the prime factorisation of 207 is: $207 = 3^2 \times 23$ Makes comparing with other numbers easier (which is $9 \times 23$ ).

✍️ FIB

Fill in the blank

What is the common prime factor of

161

and

207

?

Final Confirmation

Let's compare the factors of both numbers to see if they match the original $H C F$ and $L C M$ values.

1. Highest Common Factor (HCF)

Common prime factor: 23sharedThe only prime factor they share (appears with exponent 1 in both).
HCF = 23(Take the shared factor with lowest exponent).
✓ MatchesConfirms our answer is correct the value given in the problem.

2. Least Common Multiple (LCM) To find the LCM, we take all prime factors involved ( $3^2$ , $7$ , and $23$ Use every prime with its highest exponent):

LCM $= 3^{2} \times 7 \times 23$
LCM $= 9 \times 7 \times 23 = 63 \times 23 =$ 1449.
✓ Matches the value given in the problem.

Conclusion: Both values are verified. The answer b = 207Verified - the answer is confirmed is confirmed to be 100% correct.

💡 Tip for the Exam

Always verify your results independently.Recalculate these values to verify your work

Take a moment to calculate the HCFCheck these two values separately and LCMCheck these two values separately of your results separately (like we did with the number 207). By finding the factors independently, you can be 100% sure that your answer is solidMatching factors confirm correctness and consistent with the original data.

✍️ MCQ

Choose one

What is the primary reason for verifying the HCF and LCM independently?

Why Verify Independently?

This extra step acts as a safety net for your marksProtects you from losing points because it catches common arithmetic errorsSimple mistakes in calculations that happen during formula application.

The Verification Check: Ensure that: $\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$ Confirms your multiplication and division

If the independent check matches, you know your multiplication and division were 100% correctperfect!Your work is verified!

3. Finding LCM from HCF and Two Numbers

Let's see if you can apply the product property in a slightly different way.

The product property can also be rearranged to find LCM when you know the HCF and both numbers.
This is exactly the setup in textbook Example 7focus.

Product Property Reminder:
$\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$

✍️ Question

Problem:

Given that $HCF (252, 594) = 18$ , find $LCM (252, 594)$ .

Use the product property to solve this. Show your working.

Finding the LCM Using the Product Property

We can rearrange our "trusty" product property to solve for the LCM directly when the HCF and the two numbers are known.

The Fundamental Property

Recall that for any two positive integers $a$ and $b$ :

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

(This relationship always holds for any two positive integers)

Rearranging to solve for LCM

To isolate the LCM, we divide the product of the two numbers by their HCF:

\text{LCM} = \frac{a \times b}{\text{HCF}}

Key Formula(This key formula will save you time in board exams)

Note: This is a very handy formula to rememberMost efficient when you have HCF and both numbers but need LCM as it simplifies finding the LCM once you already have the HCF.

✍️ MCQ

Choose one

In the formula

\text{LCM} = \frac{a \times b}{\text{HCF}}

, what is the numerator?

Applying the Formula

Now, let's substitute our specific values into the formula to solve the problem.

Given:

First number ( $a$ ) = $252$ These are the two numbers we multiply together
Second number ( $b$ ) = $594$ These are the two numbers we multiply together
$HCF = < p e n a c t i o n s = " c i r c l e " c i r c l e - c o l o r = " g r e e n " c i r c l e - a n n o t a t i o n = " G i v e n " n a r r a t i o n T e x t = " t h e H C F o f 18 " c o m m e n t a r y = " T h i s i s t h e g i v e n H C F w e d i v i d e b y " > 18 < / p e n >$

Step 1: Substitute

\text{LCM} = \frac{252 \times 594}{18}

(Standard way to show your working for full marks)

We will multiply the two numbers first (or simplify by dividing) and then divide by the HCF to find the final LCM.

✍️ Yes/No

Yes or No?

Is it mathematically correct to divide

252

by

18

first, before multiplying by

594

?

Step 1: Compute $252 \times 594$

To find the LCM using the product formula, we first need to calculate the product of our two numbersFirst major step in using the formula: $252$ and $594$ .

Instead of jumping straight into long multiplication, we can use arithmetic strategiesMakes huge calculations less intimidating like breaking numbers down to make these large calculations much smoother and faster.

✍️ MCQ

Choose one

What is a 'round' number close to

594

that would make multiplication easier?

The Multiplication Trick

We can simplify the multiplication by choosing a "friendlier" number. Since $594$ is very close to $600$ , we use the distributive propertyBreak a hard number into easier subtraction:

\begin{aligned} 252 \times 594 &= 252 \times (600 - 6) \\ &= (252 \times 600) - (252 \times 6) \end{aligned}

key step(Turns long multiplication into smaller parts)

This approach allows us to solve the problem in smaller, manageable partsCan often do these in your head.

✍️ MCQ

Choose one

If

252 \times 6 = 1,512

, what is

252 \times 600

?

Final Calculation

By splitting the multiplication, we get:

Part 1: $252 \times 600 = 151, 200$ (simply multiply by $6$ and add two zerosSimple step that saves a lot of time)
Part 2: $252 \times 6 = 1, 512$

Now, subtract the second part from the first to find the total product:

252 \times 594 = 151,200 - 1,512 = \mathbf{149,688}

product(This is the product we calculated)

This result is the product of the two numbers ( $a \times b$ ), which we will later divide by the HCFFinal move to find the LCM to find the LCM.

Step 2: Divide by the HCF

To find the LCMDivide product by HCF to get LCM, we need to divide the product of the two numbers ( $149, 688$ ) by their HCFStandard exam technique for finding LCM ( $18$ ):

$\frac{149, 688}{18}$

💡 Mental Math ShortcutAvoid long division errors with this technique

Instead of performing long division with $18$ , we can break $18$ into its factors: $2$ and $9$ . Dividing in two smaller steps is often much easier:

Divide by 2:

$\frac{149,688}{2} = 74,844$
(Break eighteen into two and nine)
Divide the result by 9:
$\frac{74, 844}{9}$

This simplifies our calculation significantly!

✍️ Yes/No

Yes or No?

Is

74,844

divisible by

9

?

Chunking the DivisionBreaking division into easier pieces

Since $74, 844$ is divisible by $9$ , we can break the division into manageable chunks by finding large multiples of $9$ :

Chunk 1: We know $9 \times 8 = 72$ , so:
$9 \times 8,000 = \mathbf{72,000}$ Take out big chunks to reduce the number
Find the Remainder:
$74,844 - 72,000 = \mathbf{2,844}$ Subtract to get a smaller remainder

Now we only need to divide the remaining $2, 844$ by $9$ .

✍️ FIB

Fill in the blank

What is

9 \times 300

?

Final Calculation

Let's continue breaking down the remainder of $2, 844$ :

Chunk 2: Since $9 \times 3 = 27$ , then:
$9 \times 300 = \mathbf{2,700}$ Keep taking out chunks until done
Remainder: $2, 844 - 2, 700 = 144$
Chunk 3: From our multiplication tables:
$9 \times 16 = \mathbf{144}$ Final chunk brings remainder to zero
Remainder: $144 - 144 = \mathbf{0}$ Done!We've completely divided the number

Combine the parts:

8,000 + 300 + 16 = 8,316

(Add up all the chunks we calculated)

Result:

\mathbf{LCM = 8,316}

(Our answer from the product formula)

We have successfully found the LCM by dividing the product by the HCF!

Quick Verification: Part 1

To ensure our result is correct, let's verify it using the product property. We start by calculating the product of the HCFThese two values form the first part of the relationship and the LCMTogether with HCF, makes up the product formula:

\text{HCF} \times \text{LCM} = 18 \times 8316 = 149,688

(This product must equal the product of your original numbers)

This calculation gives us the first half of our verification check.

✍️ MCQ

Choose one

What should the product of HCF and LCM be equal to?

Quick Verification: Part 2

Now, we compare that result to the product of our two original numbers, $252$ and $594$ These are the numbers we're checking:

\text{Product of Numbers} = 252 \times 594 = 149,688

(If this doesn't match, there's an error somewhere)

Final Check:

149,688 = 149,688 \checkmark

Match!(Matching products confirm your calculation is right)

Since the product of the numbers matches the product of the HCF and LCM, our calculation is verified and correctYour safety net to confirm the answer before moving on!

💡 Alternative Calculation Shortcut

Before jumping into large multiplications, always check for a relationship between your numbers and the HCFFirst thing to look for before heavy lifting.

Observation: Notice that our HCF (18) goes into 252 exactly 14 timeskey!Huge hint to simplify the entire problem: $252 = 18 \times 14$

This is a huge hintYou can simplify the entire problem to simplify your work before multiplying!

Simplifying the Calculation

Instead of calculating the large product in the numerator first, we can substitute and cancelAlways look to simplify before multiplying:

$LCM = \frac{252 \times 594}{18}$

Substitute $252$ with $(18 \times 14)$ : $LCM = \frac{(18 \times 14) \times 594}{18}$

Cancel the 18s:(Makes the final step much faster) $LCM = 14 \times 594 = 8, 316$

This method is much faster and less error-proneCompared to long multiplication than multiplying the original large numbers.

The Golden Rule of Simplification

Using this trick helps you bypass massive, error-prone multiplicationsOften lead to silly mistakes in board exam entirely.

The Rule: Since the HCF is, by definition, a factor of the numbers, it will always divide them cleanlyH C F is a factor so it always divides cleanly.

Strategy: Whenever you are using the product formula, divide first, then multiplyMake it a habit for smoother arithmetic. It makes the arithmetic significantly smoother and more accurate.

Using the Product Formula to Find Unknown Numbers

1. Setting Up the Equation

Setting Up the Equation 📝

Problem:

Your Task ✍️

Setting Up the Problem

Step 1: Write the Product Property

The Four QuantitiesHCF, LCM, and the two original numbers

Step 2: Identify the Knowns and Unknowns

Step 3: Substitute the ValuesThe key technique in this step

Step 4: Rearrange to Isolate the UnknownSaves a lot of time in exams

💡 Key Insight: The Rearrangement Pattern

Using the Product Formula

General formula:

A Flexible Tool

Rearranging to find LCM or HCF:

2. Solving for the Missing Number

Let's complete the computation!

Your Task ✏️

Step-by-Step Multiplication: 23×144923 \times 144923×1449

Step 2: Multiplication Shortcut

Final Result

Final Step: Finding the Unknown Number bbb

The Calculation Path

Division by Estimation

Completing the Division

💡 Golden RuleA key principle for checking your work: The Whole Number Check

Verification

Factorising 207

Final Confirmation

💡 Tip for the Exam

Why Verify Independently?

3. Finding LCM from HCF and Two Numbers

Finding the LCM Using the Product Property

The Fundamental Property

Rearranging to solve for LCM

Applying the Formula

Step 1: Substitute

Step 1: Compute 252×594252 \times 594252×594

The Multiplication Trick

Final Calculation

Step 2: Divide by the HCF

💡 Mental Math ShortcutAvoid long division errors with this technique

Chunking the DivisionBreaking division into easier pieces

Final Calculation

Quick Verification: Part 1

Quick Verification: Part 2

💡 Alternative Calculation Shortcut

Simplifying the Calculation

The Golden Rule of Simplification

Step-by-Step Multiplication: $23 \times 1449$

Final Step: Finding the Unknown Number $b$

Step 1: Compute $252 \times 594$