HCF and LCM

Lesson: HCF and LCM and Solving Word Problems (Grade 6 / Prathom 6)

Welcome, Grade 6 students, to our lesson on HCF and LCM! This topic is one of the key foundations of primary school mathematics. Not only will it help you calculate much faster and master fractions later on, but it is also super useful for solving real-world problems—like packing items evenly into boxes or figuring out when events will happen at the exact same time.

If it feels a bit tricky at first, don't worry at all... Let's take it step by step. We have plenty of easy techniques and handy memory tricks ready for you!

(Helpful prerequisite knowledge to review: Factors, Prime Numbers, and Prime Factorisation from the lesson on Factors of Counting Numbers.)

Part 1: What is HCF (Highest Common Factor) and How Do We Find It?

HCF stands for the Highest Common Factor (sometimes also called the Greatest Common Divisor or GCD). It is the largest counting number that divides completely into each of the given numbers without leaving any remainder.

Imagine this: You have \(12\) orange candies and \(18\) lemon candies. You want to divide them into bags with an equal number in each bag, without mixing flavours, and you want each bag to hold the greatest possible number of candies with none left over. That greatest number of candies in each bag is precisely the HCF!

There are 3 main methods to find the HCF:

Method 1: Listing Common Factors
Best suited for smaller numbers.
Example: Find the HCF of \(12\) and \(18\)
1. Find all the factors of \(12\): \(1, 2, 3, 4, 6, 12\)
2. Find all the factors of \(18\): \(1, 2, 3, 6, 9, 18\)
3. The common factors (numbers in both lists) of \(12\) and \(18\) are \(1, 2, 3, 6\)
4. The highest common factor is \(6\)
Therefore, the HCF of \(12\) and \(18\) is \(6\).

Method 2: Prime Factorisation
Write each number as a product of its prime factors, then select only the prime factors that are shared by all the numbers and multiply them together.
Example: Find the HCF of \(12\) and \(18\)
\(12 = 2 \times 2 \times 3\)
\(18 = 2 \times 3 \times 3\)
Notice the common factors in both rows: they share one \(2\) and one \(3\).
Therefore, the HCF is \(2 \times 3 = 6\).

Method 3: Short Division (The most popular and fastest method!)
Key rule: You must find a prime number that can divide into all of the numbers completely.
Example: Find the HCF of \(24\) and \(36\)
- Divide \(24\) and \(36\) by \(2\), giving quotients of \(12\) and \(18\).
- Divide \(12\) and \(18\) by \(2\), giving quotients of \(6\) and \(9\).
- Divide \(6\) and \(9\) by \(3\), giving quotients of \(2\) and \(3\).
- Since there is no counting number greater than \(1\) that can divide both \(2\) and \(3\) evenly, stop dividing.
- Calculate the answer: Multiply all the divisors on the left side together.
Therefore, the HCF is \(2 \times 2 \times 3 = 12\).

Did you know?
When finding the HCF using short division, we only multiply the numbers on the left side going straight down, forming an "I" shape (capital letter I). We do not include the bottom quotients!

Part 2: What is LCM (Lowest Common Multiple) and How Do We Find It?

LCM stands for the Lowest Common Multiple (or Least Common Multiple). It is the smallest counting number that is a multiple of each of the given numbers. In simple terms, it is the smallest number that all the given numbers can divide into evenly.

Imagine this: One bunny hops \(4\) spaces at a time, while another bunny hops \(6\) spaces at a time. If both start from the exact same starting point, what is the first space that both bunnies will land on together? That landing spot is the LCM!

There are 3 main methods to find the LCM:

Method 1: Listing Common Multiples (Using Times Tables)
Example: Find the LCM of \(4\) and \(6\)
1. Multiples of \(4\) (\(4\)-times table): \(4, 8, 12, 16, 20, 24, 28, ...\)
2. Multiples of \(6\) (\(6\)-times table): \(6, 12, 18, 24, 30, ...\)
3. The common multiples (numbers appearing in both lists) are \(12, 24, ...\)
4. The lowest common multiple is \(12\)
Therefore, the LCM of \(4\) and \(6\) is \(12\).

Method 2: Prime Factorisation
Write each number as a product of prime factors. For any factor shared by the numbers, take it just once; then include all the remaining non-shared factors and multiply them all together.
Example: Find the LCM of \(12\) and \(18\)
\(12 = 2 \times 2 \times 3\)
\(18 = 2 \times 3 \times 3\)
- Shared factors in both rows: one \(2\) and one \(3\).
- Remaining (unshared) factors: a \(2\) from the first row and a \(3\) from the second row.
Multiply them all together: \(2 \times 3 \times 2 \times 3 = 36\)
Therefore, the LCM of \(12\) and \(18\) is \(36\).

Method 3: Short Division (Watch out—very important rules here!)
When finding the LCM of several numbers:
1. If there is a prime number that divides into at least 2 of the numbers, you can use it to divide.
2. Any number that cannot be divided evenly should simply be brought down to the next row unchanged.
3. Keep dividing until no prime number can divide into two or more numbers.
4. Calculate the answer: Multiply all the divisors on the left by all the quotients in the bottom row.

Example: Find the LCM of \(8, 12\), and \(15\)
- Divide by \(2\): \(8 \div 2 = 4\), \(12 \div 2 = 6\). Since \(15\) cannot be divided evenly by \(2\), bring down \(15\).
- Divide by \(2\): \(4 \div 2 = 2\), \(6 \div 2 = 3\). Again, bring down \(15\).
- Now we have \(2, 3, 15\). Notice that \(3\) divides both \(3\) and \(15\). Divide by \(3\): bring down \(2\), \(3 \div 3 = 1\), and \(15 \div 3 = 5\).
- The bottom row is now \(2, 1, 5\). Since no two numbers share a common factor, stop dividing.
- Calculate the answer: Multiply the divisors on the left \((2 \times 2 \times 3)\) by the numbers in the bottom row \((2 \times 1 \times 5)\).
The LCM is \(2 \times 2 \times 3 \times 2 \times 1 \times 5 = 120\).

Did you know?
When finding the LCM using short division, we multiply the numbers in an "L" shape (capital letter L)—down the left side and all the way across the bottom row!

Part 3: Summary of Differences Between HCF and LCM

HCF (Highest Common Factor):
- The final value is smaller than or equal to the smallest number in the set.
- In short division, the divisor must divide into every single number.
- Multiply only the divisors on the left side (shaped like an I).

LCM (Lowest Common Multiple):
- The final value is larger than or equal to the largest number in the set.
- In short division, the divisor can divide into at least 2 numbers (bring down any numbers that don't divide evenly).
- Multiply the divisors on the left and all numbers in the bottom row (shaped like an L).

Part 4: Solving Word Problems with HCF and LCM

When reading word problems, many students wonder: "Should I use HCF or LCM for this question?" Let's look at the clue words (keywords) that will help you decide right away!

1. Word Problems Using HCF

Clues and Keywords:
- Actions like dividing, grouping, cutting, sharing into equal parts.
- Words like "greatest", "largest", "maximum", "longest".
- "Without remainder" or dividing evenly with nothing left over.

HCF Word Problem Example:
Problem: Malee has \(24\) red roses and \(36\) white roses. She wants to arrange them into bouquets with an equal number of flowers in each bouquet, with only one colour per bouquet and no flowers left over. What is the greatest number of flowers she can put in each bouquet, and how many bouquets will she make in total?

Solution:
1. Analyse the problem: We are dividing two groups equally and want the greatest number in each bouquet, so we must find the HCF.
2. Find the HCF of \(24\) and \(36\):
- Divide \(24\) and \(36\) by \(2\) to get \(12\) and \(18\).
- Divide \(12\) and \(18\) by \(6\) to get \(2\) and \(3\).
The HCF is \(2 \times 6 = 12\).
3. Answer the questions:
- The greatest number of flowers per bouquet is \(12\) flowers.
- For the \(24\) red roses: \(24 \div 12 = 2\) bouquets.
- For the \(36\) white roses: \(36 \div 12 = 3\) bouquets.
- In total, Malee can make \(2 + 3 = 5\) bouquets.

2. Word Problems Using LCM

Clues and Keywords:
- Events that start together and repeat on their own regular cycles/intervals.
- Questions asking when things will happen "together again", "meet for the first time", "coincide".
- Words like "least", "minimum", "smallest".

LCM Word Problem Example:
Problem: Alarm clock A rings every \(15\) minutes, and alarm clock B rings every \(20\) minutes. If both clocks ring together at \(08:00\), in how many minutes will they ring together again, and at what time will that be?

Solution:
1. Analyse the problem: The events repeat in cycles, and we need to find when they will happen together next time, so we must find the LCM.
2. Find the LCM of \(15\) and \(20\):
- Divide by \(5\): \(15 \div 5 = 3\) and \(20 \div 5 = 4\).
- There are no common factors left for \(3\) and \(4\).
The LCM is \(5 \times 3 \times 4 = 60\).
3. Answer the questions:
- Both clocks will ring together again in \(60\) minutes.
- Counting forward \(60\) minutes (which is \(1\) hour) from \(08:00\), the time will be \(09:00\).

Common Mistakes (Watch Out for These!)

1. Getting Confused by the Names:
A common misconception is thinking "Highest" Common Factor means the answer must be a huge number, or "Lowest" Common Multiple means the answer must be a tiny number.
Remember: HCF is a factor (divisor), so the answer is smaller than or equal to the starting numbers. LCM is a multiple, so the answer is greater than or equal to the starting numbers!

2. Mistakes in Short Division:
- For HCF: Accidentally dividing when a number only divides into 2 values instead of all 3. Don't do this! For HCF, it must divide evenly into every single number.
- For LCM: Forgetting to bring down the numbers that do not divide evenly, or forgetting to multiply by the bottom row of quotients.

3. Answering Only One Part of a Two-Part Question:
HCF questions often ask two things: "How many items per group?" (the HCF value) AND "How many groups in total?" (adding the resulting quotients together). Always read the final question carefully!

Quick Recap

1. HCF (Highest Common Factor): Used for splitting or sharing items into equal, largest possible portions without any remainder.
- Short division rule: Must divide into all numbers.
- Multiplication rule: Multiply only the numbers on the left in an I shape.

2. LCM (Lowest Common Multiple): Used for repeating time cycles, running laps, flashing lights, or events happening together again.
- Short division rule: Can divide into at least 2 numbers (bring down the rest).
- Multiplication rule: Multiply the left side and the bottom row in an L shape.

Practice with lots of problems, keep an eye out for keywords, and you'll find that HCF and LCM can be easy and super fun to solve!