Notebook

00:04

28 Mar 2026

Real-World HCF: Tanks, Marching, and Stacking

You already know how to compute HCF.

The new challenge is modelling:

Reading a real-world problem
Recognising that the answer is 'the largest number that divides all given quantities.'

In this section, we work through three textbook examples that test this modelling skill:

Tanks: Measuring liquid capacities
Marching: Organizing groups into rows
Stacking: Arranging items efficiently

The computation is straightforward — the hard part is the translation from words to mathematics.

1. Filter — Tank/Container Problem

Let's look at a classic real-world problem that involves HCF.

The Scenario

Two milk tankers contain 504 litres and 735 litres of milk respectively.

You need to find the maximum capacity of a container that can measure the milk from either tanker an exact number of times — with no milk left over in either case.

✍️ Question

Solve this problem completely:

(a) Explain why this is an HCF problem.
(b) Compute the answer (the maximum container capacity).
(c) State how many times the container fills each tank.

Modeling the Problem

Let me show you how to model this problem. Modeling is essentially translating the words of the problem into a mathematical planTranslating words into a mathematical plan.

Turning Story into Math

The Context: We have two milk tankers that need to be emptied.
The Goal: We need to build a solution by identifying the mathematical principles hidden in the story.

Decoding the Language

What does 'measure the milk an exact number of times' mean?

In word problems, this phrase is a very important clue. It tells us something specific about the relationship between the measuring containerThe key tool we use to solve the problem and the tanker.

Think of it like using a measuring cup to empty a big bucket—we want it to be a perfect fit without any leftover liquid.A perfect fit with nothing left over

✍️ Yes/No

Yes or No?

If you use a

2

liter jug to empty a

7

liter tank, will it be measured an 'exact number of times'?

The Meaning of 'Exact'

It means the container fills up completely, is emptied into something, fills up again, is emptied again, ... and at the end, the tanker is completely emptygoal(The container size is a factor of the tanker).

The container must fill to the very top every single time.
No milk is left over; the tanker is left completely dry.
If even a tiny drop remains that doesn't fill the container, it is not an exact number of times.

Key takeaway: In mathematical terms, 'exact number of times' means there is zero remainderSimply means there is zero remainder.

✍️ MCQ

Choose one

If a container measures a tanker's capacity an 'exact number of times', the container's capacity must be a _______ of the tanker's capacity.

Modeling the Math

Let's turn the milk measurement into a mathematical plan. If our container has a capacity of $C$ litres and we are emptying a tanker with $504$ litres, here is how the remaining volume changes:

After 1 filling: $504 - C$ litres remain.
After 2 fillings: $504 - 2 C$ litres remain.
After $k$ fillings: $504 - k C$ litres remain.

Achieving an "Exact Measurement"

For a measurement to be exact, the tanker must be emptied completely with zero remainderZero remainder after pouring. Mathematically, this means:

$504 - k C = 0$

\text{or simply: } k \times C = 504

(Container size fits into total volume without any left over)

This shows that $504$ is perfectly divisibleContainer size fits into total volume without any left over by $C$ .

✍️ Yes/No

Yes or No?

k \times C = 504

, is

C

a factor of

504

Both Tankers Must be Measured

We have two tankers to consider. The same logic we used for the first tanker applies to the second one containing $735$ litres.

To empty the first tanker perfectly, $C$ must be a factor of $504$ Capacity must divide 504 exactly.
To empty the second tanker perfectly, $C$ must also be a factor of $735$ Capacity must also divide 735 exactly.

The Common Factor

Since a single container $C$ must work for both tankers, we can conclude: $C$ is a common factor of both $504$ and $735$ .Satisfies both conditions simultaneously

✍️ MCQ

Choose one

To find the maximum capacity, which should we calculate?

The Goal: HCF

To save time and be most efficient, we need the MAXIMUMYour clue to find the Highest Common Factor possible capacity for our container $C$ .

In mathematics, the largest number that divides two or more numbers exactly is the Highest Common Factor (HCF)The largest number that divides both quantities exactly.

Our Math Task:

\text{Maximum Capacity} = HCF(504, 735)

Goal(Divide two quantities into largest possible equal parts)

Isn't it amazing how a real-world problem like "measuring milk" boils down to finding a simple HCFDivide two quantities into largest possible equal parts?

Step 1: Prime Factorisation of 504

To find the HCF, let's first break down 504 into its prime building blocks using the factor tree or division method:

$504 = 2 \times 2 \times 2 \times 3 \times 3 \times 7$

Writing with exponents:

504 = 2^3 \times 3^2 \times 7^1

(Always express large numbers using powers)

We have three 2sTwo cubed means three twos multiplied, two 3sThree squared means two threes multiplied, and one 7.

✍️ MCQ

Choose one

What is the prime factorisation of

735

Step 2: Prime Factorisation of 735

Next, we perform the prime factorisation for 735:

$735 = 3 \times 5 \times 7 \times 7 = 3^{1} \times 5^{1} \times 7^{2}$

Identifying Common Factors

To find the HCF, we identify the prime numbers that appear in both lists and select the lowest powerOnly take the lowest exponent of each common factor for each:

Common Primes:
- 3: Comparing $3^{2}$ and $3^{1} \to$ Minimum power is 1.
- 7: Comparing $7^{1}$ and $7^{2} \to$ Minimum power is 1.

Note: Even though 2 and 5 are factors, they are not common to both numbersFactors must appear in both numbers to count.

✍️ MCQ

Choose one

The HCF is the product of the lowest powers of common factors. Which calculation is correct?

The Result

By multiplying these common factors with their lowest powers, we find the maximum capacity required:

HCF $(504, 735) = 3^{1} \times 7^{1} =$ $\mathbf{21}$ answerTwenty-one goes into both numbers exactly

Final Answer

The maximum capacity of the container is 21 litres(Word problems use this phrase as a cue for HCF).

This 21-litre container is the largest size that can measure the milk in both tankers an exact number of timesThe HCF is the biggest measure that works for both.

Interpretation: What does 21 litres actually mean?

We calculated the HCF as 21. In the context of our problem, this represents the maximum capacity of the containerThe largest possible size that fits into both volumes.

Why 21 litres?

The Largest Size: It is the largest possible capacity that divides both tanker volumes ( $504$ and $735$ ) perfectly.
Exact Measurement:No milk left over at the end This container will measure the milk from both tankers an exact number of times.

Note: Using this size ensures there is no milk left over and no partial fillings required. It is "just right!"

✍️ MCQ

Choose one

How many times does a

21

-litre container go into

504

litres?

The Practical Meaning

To see the HCF in action, let's look at how many times the $21$ -litre container is used for each tanker:

First tanker ( $504$ litres): $504 \div 21 = 24$ fillings
Second tanker ( $735$ litres): $735 \div 21 = 35$ fillings

💡 Exam Tip: Don't Stop at the Number!Remember this when solving in your exam

To ensure you get full marks in your exam:

Always include units:Shows you understand the physical quantity Write " $21$ litres", not just " $21$ ".
Explain the meaning:Explain the meaning clearly State clearly that it represents the "maximum capacity" or the "number of fillings" required.

2. Marching Problem — Indirect Phrasing

New Problem 🎖️

This one uses slightly different language — instead of saying 'divide equally,' it talks about 'same number of columns.'

But the underlying structure is the same as what you've seen before.

Let's see if you can crack it!

📋 Given Info

The Problem

An army contingent of 612 members is to march behind a band of 48 members in a parade.

The two groups must march in the same number of columns.

What is the maximum number of columns in which they can march?

✍️ Question

Solve completely:

(a) Explain what same number of columns requires mathematically.
(b) Why is this an HCF problem?
(c) Find the answer and interpret it (how many columns? how many rows for each group?).

Translating the Problem into Mathematics

Let's visualize the "same number of columns"The width of both groups must be identical requirement:

The Formation: Imagine the parade ground. Both the army and the band march in solid rectangular formationsA way to divide total people into equal rows and columns.
The Structure: Each formation has a specific number of columns (width) and rows (depth).

To march together, the width (number of columns) must be identical for both groups.

✍️ FIB

Fill in the blank

If you have

20

people and you arrange them into

4

columns, how many rows will you have?

The Condition for the Army

Let $c$ represent the number of columns.

Army Total: $612$ It must divide the total exactly with no remainder members.
Formation: Arranged in $c$ columns, the number of rows is $\frac{612}{c}$ .

Key Constraint: Since we cannot have a fraction of a personYou cannot have a fraction of a person! To have full rows, $c$ must be a factor of $612$ Must divide the total exactly with no remainder (it must divide $612$ perfectly).

✍️ MCQ

Choose one

If the band has

48

members and uses the same number of columns

c

, what must be true about

c

and

48

Finding the Maximum Columns

Applying the same logic to the band:

Band Total: $48$ members.
Condition: In $c$ columns, there are $\frac{48}{c}$ rows. For this to work, $c$ must divide $48$ .

The Mathematical Goal

Since $c$ must divide both numbers, $c$ is a common factorThe columns must work for both groups of $612$ and $48$ . Because the problem asks for the maximumA direct signal to calculate the HCF number of columns possible, we need to find the largest such value:

Maximum columns = $\text{HCF}(612, 48)$ answerA direct signal to calculate the HCF

Visualising the two formations side by side makes it clear why both groups need the same column count — and why that column count must divide both 612 and 48 exactly.

✍️ MCQ

Choose one

For the number of rows to be a whole number in BOTH formations,

c

must divide evenly into:

Finding the Maximum Columns

To find the maximum number of columns in which both groups can march, we need to calculate the Highest Common Factor (HCF)Maximum in equal groups means find HCF of 612 and 48.

Step 1: Prime Factorisation of 612

Let's break down 612 into its prime building blocks:

$612 \div 2 = 306$
$306 \div 2 = 153$
$153 \div 3 = 51$
$51 \div 3 = 17$ (prime)(Prime number)17 is prime, cannot be divided further

Prime Factorisation: $612 = 2^{2} \times 3^{2} \times 17$

✍️ FIB

Fill in the blank

In the prime factorisation of

48

, how many times does the factor

2

appear?

Step 2: Prime Factorisation of 48

Now, let's look at the band members. We divide 48 by 2 repeatedly until we are left with a single 3:

$48 \div 2 = 24$
$24 \div 2 = 12$
$12 \div 2 = 6$
$6 \div 2 = 3$
$3 \div 3 = 1$

Prime Factorisation:

48 = 2^4 \times 3^1

✍️ MCQ

Choose one

To find the HCF from prime factorisations, we take the product of the _____ power of each common prime factor.

Step 3: Identify Common Factors & Calculate HCF

To ensure the result divides into both numbers, we follow this rule:

Rule: For HCF, take the lowest power of each common prime factor.HCF uses lowest power of common primes

Common Prime 2: Powers are $2^{2}$ and $2^{4}$ . The minimum is $2^2$ minChose 2 squared as smallest exponent.
Common Prime 3: Powers are $3^{2}$ and $3^{1}$ . The minimum is $3^1$ min(3 to the power 1 is the smaller exponent).

Calculation: $HCF = 2^{2} \times 3 = 4 \times 3 = 12$

Result: The maximum number of columns is 1217 wasn't common to both, so we ignored it.

What does the HCF tell us?

We found that the $HCF (612, 48) = 12$ . In the context of the parade, this represents the maximum number of columnsThis is what the problem is really asking for both groups can march in while remaining perfectly aligned.

Formation Breakdown:

Army Contingent: $\frac{612}{12} = 51$ rows
Army Band: $\frac{48}{12} = 4$ rows

The Result: Because $12$ is a common factorOnly way both groups get whole number of rows, both groups finish with a whole number of rowsNo one gets left behind in the formation, creating a perfect rectangular formation for the parade.

✍️ Yes/No

Yes or No?

If the number of columns is not a factor of the total members (e.g.,

10

columns for

48

members), will the rows be whole numbers?

Why not use LCM?

It can be tempting to use LCM in word problems, but let's see why it doesn't work here:

$LCM (612, 48) = \frac{612 \times 48}{12} = 2448$

Does "2448 columns" make sense? No. You cannot arrange $48$ band members into $2448$ columns! You can never have more columns than the total number of people.You cannot have more columns than people to arrange

Rule of Thumb

Use HCFUse HCF when dividing into largest equal groups when you need to divide or split items into the largest possible equal groupsThe biggest groups where everything divides evenly.
Use LCM when you need to find a future point where repeating events overlap or coincide.

✍️ MCQ

Choose one

When a problem requires dividing quantities into the largest possible equal groups, we calculate the: