Welcome! Today we're exploring Coprime Pairs, Simplest Form, and Validating HCF-LCM Pairs — three connected ideas that will make your number theory work much smoother.

Some pairs of numbers share no prime factors at all — like:

Pair	Prime factors
8 and 15	2³ vs 3×5
25 and 36	5² vs 2²×3²

These are called coprime pairs.

Coprime pairs have special properties:

Their HCF is 1always, and their LCM is just their product.

But coprime pairs also power two practical skills you'll use constantly:

Reducing fractions to simplest form — essential before checking if a decimal terminates
A five-second test — to check if a given HCF and LCM can actually belong to some pair of numbers

By the end of this section, you'll handle all three:

✓ Identify coprime pairs
✓ Reduce fractions to simplest form
✓ Validate HCF-LCM pairs instantly

1. Coprime numbers and their key property

Before we dive into simplest form and validation, you need to understand coprime pairs — what they are and why they matter.

Their defining property is the foundation for everything else in this section.

📋 Given Info

Two numbers are coprime (also called relatively prime) if their HCF is 1 — they share no common prime factor.

Example:

$8 = 2^{3}$
$15 = 3 \times 5$

Since 8 and 15 share no prime factor, they are coprimecoprime!.

✍️ Question

Your turn! 🤔

Are 14 and 35 coprime?
Are 25 and 36 coprime?

For each pair, explain your reasoning using prime factorisations.

Then state: If two numbers are coprime, what is their LCM equal to?

CoprimeTwo numbers share absolutely nothing means HCF = 1No prime divides both of them — no shared prime factorThey share absolutely nothing between the two numbers.

In simple words: Two numbers are coprime (also called relatively prime) when they have nothing in common except 1.

Think of it this way — if you break both numbers into their prime factors, you won't find any prime that appears in both.

✍️ Yes/No

Yes or No?

We said

8 = 2^3

and

15 = 3 \times 5

. Do they share any common prime factor?

✓

Key Point: Coprime = HCF is 1 = No common prime factorAll the same thing in exams

All three statements mean the same thing!same!Prove any one and you've proved all three

✍️ FIB

Fill in the blank

If two numbers are coprime, their HCF is ___.

✓

Check 14 and 35:

$14 = 2 \times 7$

$35 = 5 \times 7$

✍️ MCQ

Choose one

Looking at the prime factorisations

14 = 2 \times 7

and

35 = 5 \times 7

, which prime factor do they share?

✓

Shared prime: 7commonFinding a shared prime means they're not coprime. So $HCF (14, 35) = 7$ , which is not 1HCF must equal 1 for coprimality. They are NOT coprimeThe test confirms they share a factor.

Check 25 and 36:

$25 = 5^2$

$36 = 2^2 \times 3^2$

✍️ Yes/No

Yes or No?

Looking at

25 = 5^2

and

36 = 2^2 \times 3^2

, do these two numbers share any common prime factor?

✓

Shared primes: noneNo shared primes between them. The primes of 25 are ${5}$ and the primes of 36 are ${2, 3}$ . No overlap. So $\text{HCF}(25, 36) = 1$ HCF=1When no overlap, HCF must be one. They ARE coprimeThe test for coprime numbers.

✍️ MCQ

Choose one

If two numbers are coprime, their LCM equals:

✓

The key property of coprimeWhen coprime, HCF equals 1 pairs: Since HCF = 1, the product propertyThe formula becomes much simpler gives:

$a \times b = HCF \times LCM = 1 \times LCM = LCM$

So for coprime numbers, the LCM is simply their productYour shortcut for coprime pairs.

✍️ MCQ

Choose one

If two numbers are coprime, their LCM equals:

✓

For example, $LCM (8, 15) = 8 \times 15 = 120$ , because 8 and 15 are coprime.

This is a useful shortcut: whenever you spot that two numbers share no prime factorsNo common factors means they're coprime, you know their LCM immediatelyJust multiply them directly without any computation.

✍️ FIB

Fill in the blank

We found that

25

and

36

are coprime. What is

\text{LCM}(25, 36)

?

✓

2. Reducing a fraction to simplest form

Let's check your understanding of reducing fractions to simplest form. 📝

A fraction is in simplest form when the numerator and denominator are coprime — meaning they share no common factors other than 1.

To reduce a fraction to simplest form:

Find the HCF of the numerator and denominator
Divide both by that HCF

This skill becomes essential when you later need to classify decimals as terminating or non-terminating.

✍️ Question

Your Task ✏️

Reduce $\frac{234}{390}$ to simplest form.

Show:

The HCF computation (using prime factorisation)
The final reduced fraction
How do you confirm the result is truly in simplest formverify?

To reduce $\frac{234}{390}$ , we need to find the HCF and divide both the numerator and denominator by it.

Remember: A fraction is in simplest form when the numerator and denominator are coprimeThey share no common factors at all — that is, their HCF equals 1HCF equals exactly 1.

Prime Factorise:

$234 = 2 \times 117 = 2 \times 9 \times 13 = 2 \times 3^2 \times 13$
$390 = 2 \times 195 = 2 \times 3 \times 65 = 2 \times 3 \times 5 \times 13$

So we have:

$234 = 2^1 \times 3^2 \times 13^1$
$390 = 2^1 \times 3^1 \times 5^1 \times 13^1$

✍️ MCQ

Choose one

What is

\text{HCF}(234, 390)

?

✓

Common primesOnly look at primes shared by both numbers at lowest powersAlways choose the smaller exponent for HCF:

2: $\min (1, 1) = 1 \to 2^{1}$
3: $\min (2, 1) = 1 \to 3^{1}$
13: $\min (1, 1) = 1 \to 13^{1}$

So, $\text{HCF}(234, 390) = 2 \times 3 \times 13 = 78$ The largest number that divides both evenly.

✍️ MCQ

Choose one

What is

\frac{234}{78}

?

✓

Divide both by HCF (78)This is the method for reducing fractions:

$234 \div 78 = 3$
$390 \div 78 = 5$

Simplified fraction: $\frac{3}{5}$ The result after dividing by HCF

Confirmation: $HCF (3, 5) =$ $1$ The test to verify fully reduced

Since 3 and 5 are different primes, they share no common factors. So $\frac{3}{5}$ Coprime numbers share no common factors is in simplest formDefinition of coprime numbers.

✍️ MCQ

Choose one

Which of the following fractions is already in simplest form?

✓

Why does this matter?

Later, when checking if a decimal terminates, you MUST reduce the fraction firstThis is non-negotiable when checking terminating decimals.

The rule about denominator factors (only 2s and 5sApplies to the simplified form, not the original) applies to the SIMPLIFIEDkeyThe denominator after reducing to simplest form denominator, not the original one.

Skipping this step can give the wrong answer.Always reduce first, then check the factors

3. The HCF-divides-LCM validation test

Here's a quick filter that catches impossible HCF-LCM pairs in seconds. It comes up in multiple-choice questions and can save you from chasing a nonexistent answer.

Key fact:

For any valid pair of numbers, the HCF always divides the LCM exactly.

Why?

Every prime power in the HCF is the minimum of the two numbers' powers
Every prime power in the LCM is the maximum of the two numbers' powers
Since $\min \leq \max$ for every prime, the HCF is always a factor of the LCM

If $HCF ∤ LCM$ , the pair is impossibleinvalid.

✍️ Question

Question 🤔

Is it possible to have two numbers whose HCF is 18 and LCM is 760?

Check using the divisibility test
Then explain in one sentence WHY HCF must always divide LCM

The test: does HCF divide LCM exactlyThis division tells you if the pair works?

$760 \div 18 = 42.222...$ Not a whole number means invalid pair

(not whole)42.222...Decimal result proves invalidity

Not exact. So 18 does NOTHCF must divide LCM with no remainder divide 760.

This means no pair of numbers can have HCF = 18 and LCM = 760. Such a pair is impossibleNo two numbers can have this HCF and LCM.

Why must HCF always divide LCM? Every prime power in the HCF is the minimumHCF uses the smaller exponent from each number of the two numbers' powers, while the LCM takes the maximumLCM uses the larger exponent from each number. Since min ≤ max always, the HCF is automatically a factor of the LCMMinimum is always less than or equal to maximum.

✍️ Yes/No

Yes or No?

Can two numbers have HCF

= 12

and LCM

= 90

?

✓

Why must HCF always divide LCMThis follows from how prime powers work? Think about it prime by prime.

For any prime $p$ , suppose the two numbers have powers $a$ and $b$ (where $a \leq b$ ). Then:

HCF gets $p^a$ HCF picks the smaller exponent (the minmin(The smaller of the two powers))
LCM gets $p^b$ LCM picks the larger exponent (the maxmaxThe larger of the two powers)

Since $a \leq b$ , we have $p^a$ divides $p^b$ Because a is less than or equal to b. This holds for every primeThis divisibility applies to all primes. So the HCF divides the LCMThat's why the divisibility test works.

✍️ MCQ

Choose one

If for a prime

p

, one number has

p^2

and another has

p^5

, which power goes to the HCF and which to the LCM?

✓

In the example: $18 = 2 \times 3^2$ and $760 = 2^3 \times 5 \times 19$ .

For HCF to be 18, the two numbers must both have at least $3^2$ HCF needs this power in both numbers. But for LCM to be 760, the highest power of 3 in either number would be $3^0$ LCM reflects what's in at least one number (since 760 has no factor of 3No factor of 3 means max power is zero).

✍️ MCQ

Choose one

If the HCF contains

3^2

, what does that tell us about the power of 3 in BOTH original numbers?

✓

You can't have min = 2Min power can't be more than max and max = 0 for the same prime — contradictionimpossible!This is mathematically impossible.

This is why the divisibility testQuick test catches this contradiction works: if HCF doesn't divide LCM, there's a prime where the "minimum" exceeds the "maximum"Found a prime where min exceeds max — which is mathematically impossible.

✍️ Yes/No

Yes or No?

Can two numbers have HCF = 12 and LCM = 60?

✓

Quick Tip: This five-second divisibility checkYour first step before any calculation saves minutes of fruitless computationStop right there if it doesn't divide evenly.

4. Why simplifying fractions matters for later topics

You can reduce fractions and validate HCF-LCM pairs. The last piece is understanding why simplest form is not optional — it's a mandatory prerequisite for the decimal classification that comes later.

Here's an important rule you'll use:

The Terminating Decimal Rule: A fraction (in simplest form) has a terminating decimal if and only if the denominator's prime factorisation contains only 2s and 5s.

✍️ Question

Consider the fraction $\frac{6}{15}$ .

If you check $\frac{6}{15}$ without simplifying, the denominator is $15 = 3 \times 5$ , which has a factor of 3.

Would you conclude the decimal doesn't terminate?

Now simplify and check again.

What do you get, and what's the lesson?

Let's trace both paths for $\frac{6}{15}$ .

Path 1 — Without simplifying:

Denominator = $15 = 3 \times 5$ .

The termination rule says: only 2s and 5s allowedThe key rule for terminating decimals. The denominator has a 3.

Verdict: non-terminating. ❌ (This is WRONG!)

Path 2 — After simplifying:

HCF(6, 15): $6 = 2 \times 3$ Break down into prime factors, $15 = 3 \times 5$ Break down into prime factors. Common prime: 3commonThe shared factor between both. So HCF = 3Use this to reduce the fraction.

$\frac{6}{15} = \frac{6 \div 3}{15 \div 3} = \frac{2}{5}$

New denominator = $5 = 5^{1}$ .

Only 5s — no other primes. Verdict: terminating. ✓

And indeed, $\frac{2}{5} = 0.4$ .

The lesson: The terminating decimal rule ONLY works on simplified fractionsMust simplify before applying the rule. The 3 in the original denominator was a "fake" obstructionIt disappears when you simplify — it cancelled out!

✍️ MCQ

Choose one

Why did checking

\frac{6}{15}

without simplifying give the wrong conclusion about termination?

✓

Path 1 gave the WRONG answer. The fraction $\frac{6}{15}$ does terminate, but checking the unsimplified denominator said it doesn't.

This is a critical lesson — the termination rule only works on simplified fractionsMust simplify before checking denominators. If you skip simplification, you can get completely wrong conclusionsWrong answers from unsimplified fractions!

What went wrong? The factor of 3 in the denominator was 'cancelled' by the factor of 3 in the numerator. Simplifying removes these shared factors, changing the effective denominator from 15 to 5.

The 3 was a fake obstructionkey insightIt cancelled out, never actually blocking termination — it looked like a problem, but it wasn't really there in the simplified form.

✍️ MCQ

Choose one

Consider the fraction

\frac{21}{35}

. Without simplifying, the denominator is

35 = 5 \times 7

. After simplifying, what is the denominator?

✓

The lesson: The termination rule applies to the SIMPLIFIED denominatorCheck the simplified form, not the original, not the original. Always reduce to simplest form firstAlways simplify before checking for 2s and 5s.

In some cases (like $\frac{6}{15}$ exampleSkipping this step can flip your answer), skipping this step flips the answer completelyYou might say non-terminating when it's actually terminating.

✍️ MCQ

Choose one

The fraction

\frac{14}{35}

has denominator

35 = 5 \times 7

. After simplifying, what is the new denominator?

✓