Before we can find the highest common factor or the least common multiple, we need to be clear on what factors and multiples are. They're related, but they describe different things.

Think of factors as the building blocks of a number. They are the integers you can multiply together to get that number. For example, to get 12, you could multiply $2 \times 6$ or $3 \times 4$. So, 2, 6, 3, and 4 are all factors of 12.

Multiples are what you get when you skip-count by a certain number. The multiples of 12 are 12 (which is $12 \times 1$), 24 ($12 \times 2$), 36 ($12 \times 3$), and so on, stretching out to infinity.

The Highest Common Factor (HCF) is the largest number that is a factor of two or more other numbers. It's also known as the Greatest Common Divisor (GCD). The HCF helps us find the biggest possible piece we can divide things into evenly.

One way to find the HCF is by using prime factorization. This means breaking each number down into its prime factors—the prime numbers that multiply to give the original number.

For larger numbers, the division method is often faster. To find the HCF of two numbers, you divide the larger number by the smaller one. Then you take the remainder and divide the previous divisor by it. You repeat this until the remainder is zero. The last non-zero divisor is your HCF.

Let's find the HCF of 198 and 360.

1. Divide 360 by 198. You get a quotient of 1 and a remainder of 162.
2. Now divide the previous divisor (198) by the remainder (162). You get a quotient of 1 and a remainder of 36.
3. Divide the previous divisor (162) by the new remainder (36). You get a quotient of 4 and a remainder of 18.
4. Divide 36 by 18. You get a quotient of 2 and a remainder of 0. Since we reached a remainder of 0, we stop. The last divisor we used was 18, so the HCF of 198 and 360 is 18.

The Least Common Multiple (LCM) is the smallest number that is a multiple of two or more other numbers. It's useful when you need to sync up events that happen in cycles.

We can use prime factorization for the LCM, too. After breaking each number down into its prime factors, you take the highest power of every prime factor present in any of the numbers and multiply them together.

Let's find the LCM of 18 ($2 \times 3^2$) and 24 ($2^3 \times 3$).

1. The prime factors involved are 2 and 3.
2. The highest power of 2 is $2^3$ (from 24).
3. The highest power of 3 is $3^2$ (from 18).
4. Multiply these highest powers together: $2^3 \times 3^2 = 8 \times 9 = 72$.

So, the LCM of 18 and 24 is 72.

The division method for LCM is also very efficient, especially for more than two numbers. Let's find the LCM of 15, 20, and 30.

1. Write the numbers in a row: 15, 20, 30.
2. Find a prime number that divides at least two of them. Let's use 5. Divide the numbers by 5, writing the quotients below. If a number isn't divisible, just bring it down. We get 3, 4, 6.
3. Now we have 3, 4, 6. Let's use the prime number 2. We get 3, 2, 3.
4. Now we have 3, 2, 3. Let's use the prime number 3. We get 1, 2, 1.
5. We stop when no two numbers share a prime factor.

To find the LCM, multiply all the divisors and the remaining numbers: $5 \times 2 \times 3 \times 1 \times 2 \times 1 = 60$.

There's a cool relationship between the HCF and LCM of any two positive integers, let's call them a and b. The product of the two numbers is always equal to the product of their HCF and LCM.

This means if you've already calculated the HCF, you can find the LCM quickly without going through the whole process again. For our earlier example with 18 and 24, we found the HCF was 6.

Using the formula: $18 \times 24 = 6 \times LCM(18, 24)$ $432 = 6 \times LCM(18, 24)$ $LCM(18, 24) = 432 \div 6 = 72$

It works every time and can be a real time-saver.

Time to check what you've learned.

Understanding HCF and LCM is a key step in mastering how numbers relate to each other. They're not just abstract concepts; they're practical tools for simplifying fractions, scheduling events, and solving a wide range of problems.