Some problems give you HCF and LCM values and ask whether they are possible.

For example: "Can two numbers have $H C F = 18$ and $L C M = 760$ ?"

Before trying to find the numbers (which would fail), you should check a simple condition:

Does the HCF divide the LCM?
If not, the pair is impossible.

This section teaches you:

The quick test for validity
The mathematical reasoning behind it
How to justify your answer

1. The Divisibility Condition

Determining if an HCF-LCM Pair is Possible 🔍

Some problems give you HCF and LCM values and ask whether they are possible.

Example: "Can two numbers have $H C F = 18$ and $L C M = 760$ ?"

Before trying to find the numbers (which might fail), there is a key test for validity:

Does the HCF divide the LCM?

Let's see if you can apply this test and explain why it must hold.

✍️ Question

Your Turn ✏️

Claimed pair:

HCF = $18$
LCM = $760$

The Question: Can two numbers have $H C F = 18$ and $L C M = 760$ ?

Check by testing whether HCF divides LCM.

Explain why this test works.

The key test: does HCF divide LCM?

Let's check: $760 \div 18 = 42.222...$

✍️ Yes/No

Yes or No?

Is the result of

760 \div 18

a whole number?

This is NOT a whole number. So 18 does not divide 760.

Conclusion: No two numbers can have HCF = 18 and LCM = 760.

Why must HCF divide LCM?

Think about what these terms mean:

HCF = 18 means both numbers are divisible by 18.
We can write the two numbers as: $a = 18 m$ and $b = 18 n$ for some positive integers $m$ and $n$ .

✍️ Yes/No

Yes or No?

If

a = 18m

, is

a

a multiple of

18

?

The Logic Chain

LCM is the smallest number divisible by both $a$ and $b$ .
Since $a = 18 m$ , any multiple of $a$ is also a multiple of 18.
The LCM is a common multiple of $a$ and $b$ , so it's definitely a multiple of $a$ .

✍️ T/F

True or False?

If

x

is a multiple of

10

, and

y

is a multiple of

x

, is

y

a multiple of

10

?

The Final Conclusion

LCM is a multiple of $a$ , and $a$ is a multiple of 18. Therefore, LCM is a multiple of 18.

But we know that 760 is NOT a multiple of 18. This is a contradiction.

Conclusion: No such pair of numbers exists.

General rule: For any valid HCF-LCM pair, the LCM must be a multiple of the HCF. Equivalently, HCF must divide LCM. If this fails, the pair is impossible.

✍️ MCQ

Choose one

If the HCF of two numbers is

5

, which of these could be their LCM?

Quick way to check: Just divide LCM by HCF. If you get a whole number, the pair MIGHT be valid. If you get a decimal, the pair is definitely invalid.

✍️ T/F

True or False?

If dividing the LCM by the HCF results in

4.5

, the HCF-LCM pair is valid.

2. Applying the Validity Test to Multiple Cases

📋 Given Info

Let's practise the validity test on several cases, including a tricky one where HCF equals LCM.

Here are three claimed HCF-LCM pairs:

(a) HCF = $12$ , LCM = $180$
(b) HCF = $25$ , LCM = $600$
(c) HCF = $9$ , LCM = $9$

✍️ Question

Your Task 🔍

For each pair, determine if two numbers CAN have the given HCF and LCM.

Show the divisibility check for each:

(a) HCF = $12$ , LCM = $180$
(b) HCF = $25$ , LCM = $600$
(c) HCF = $9$ , LCM = $9$

Case (a): HCF = 12, LCM = 180

Let's apply our division test to this first pair:

$180 \div 12 = 15$

Is 15 a whole number? Yes.

✍️ Yes/No

Yes or No?

Since

180 \div 12 = 15

, is this HCF-LCM pair valid?

Because the division gives a whole number, this pair is valid.

Example numbers: To prove it, let's look at two numbers that fit this:

$a = 12 \times 3 = 36$
$b = 12 \times 5 = 60$

If you check these: $HCF (36, 60) = 12$ and $LCM (36, 60) = 180$ . It works perfectly!

✍️ MCQ

Choose one

If

\text{LCM} \div \text{HCF} = 15.5

, is the pair possible?

(b) HCF = 25, LCM = 600

Let's apply our rule to a new set of numbers. This time, we have an HCF of 25 and an LCM of 600.

✍️ MCQ

Choose one

To test the validity of the HCF-LCM pair, what should we do?

$600 \div 25 = 24$

Since 24 is a whole number, this pair is valid.

✍️ Yes/No

Yes or No?

Is

25

a factor of

600

?

Example numbers: $a = 25$ , $b = 600$ .

Check:

HCF(25, 600) = 25
LCM(25, 600) = 600

It works!

(c) HCF = 9, LCM = 9

Let's test this pair using our rule:

$9 \div 9 = 1$

Since $1$ is a whole number, the condition is satisfied.

✍️ Yes/No

Yes or No?

Is

1

a whole number?

So this pair is valid.

This might surprise you: Can HCF and LCM be EQUAL?

Yes! This happens specifically when both numbers are the same:

Let $a = 9$ and $b = 9$ .
HCF(9, 9) = 9
LCM(9, 9) = 9

✍️ FIB

Fill in the blank

If

a = 15

and

b = 15

, then

\text{LCM}(15, 15) = \text{?}

The $a = b$ Condition

In general, HCF = LCM if and only if $a = b$ .

The HCF is always $\leq$ LCM (since HCF divides LCM), and they are equal only when both numbers coincide.

✍️ Yes/No

Yes or No?

If

a = 12

and

b = 18

, can the HCF be equal to the LCM?

The Complete Picture

To summarize everything we've learned, here is the checklist for a valid HCF-LCM pair:

HCF $\leq$ LCM always.
HCF divides LCM always.
HCF = LCM only when $a = b$ .

✍️ T/F

True or False?

The HCF of any two positive integers can never be greater than their LCM.

Identifying Invalid Pairs

If any of these conditions fail, the pair is invalid.

Whether it's the division test or the size comparison, if the rules are broken, those two numbers simply cannot exist.

3. Writing a Complete Board Exam Answer

Board Exam Practice 📝

For the exam, you need to write a clear, well-structured answer.

Let's practise with the exact question from Ex 1B Q6:

"Is it possible to have two numbers whose HCF is 18 and LCM is 760? Give reasons."

✍️ Question

Ex 1B Q6:

Can two numbers have $18$ as their HCF and $760$ as their LCM? Give reasons.

Write a complete answer to this question as you would in a board exam.

Your response should include:

The general rule we discussed.
The specific check for these numbers.
A clear conclusion based on your findings.

Model Exam Answer

Question: Can two numbers have 18 as their HCF and 760 as their LCM? Give reasons.

Answer:

No, two numbers cannot have HCF = 18 and LCM = 760.

Reason: For any two positive integers $a$ and $b$ , the HCF must always divide the LCM.

✍️ MCQ

Choose one

What calculation checks if

\text{HCF}=20

and

\text{LCM}=500

is possible?

Why must HCF divide LCM?

HCF divides both $a$ and $b$ (by definition of HCF).
LCM is a common multiple of $a$ and $b$ (by definition of LCM), so LCM is divisible by $a$ .
Since $a$ is divisible by HCF, and LCM is divisible by $a$ , it follows that LCM is divisible by HCF.

Check: Does 18 divide 760?

$760 \div 18 = 42.22...$

(not a whole number)

Conclusion: Since 18 does not divide 760, the LCM cannot be 760 when the HCF is 18. Therefore, no such pair of numbers exists.

Structure for full marks:

State the rule: HCF must divide LCM.
Give the reason: Both numbers are multiples of HCF, so LCM is too.

Do the check: Show $760 \div 18$ is not a whole number.
State the conclusion: The pair is impossible.

This format works for any similar question on the exam.

Determining if an HCF-LCM Pair is Possible

1. The Divisibility Condition

Determining if an HCF-LCM Pair is Possible 🔍

Why must HCF divide LCM?

The Logic Chain

The Final Conclusion

2. Applying the Validity Test to Multiple Cases

Your Task 🔍

Case (a): HCF = 12, LCM = 180

(b) HCF = 25, LCM = 600

(c) HCF = 9, LCM = 9

The a=ba = ba=b Condition

The Complete Picture

Identifying Invalid Pairs

3. Writing a Complete Board Exam Answer

Board Exam Practice 📝

Model Exam Answer

Why must HCF divide LCM?

The $a = b$ Condition