Highest Common Multiple: 2 Methods To Solve HCF Word Problems

Introduction

Let’s do a quick recap on the Highest Common Multiple.

🔎 What Is HCF? 🔎

The Highest Common Factor (HCF) is the largest number that can divide given numbers without leaving a remainder.

In this blog post, I will be sharing with you two methods you can use to find the HCF quickly and accurately — the Prime Factorisation and Ladder Methods!

Let’s Take A Look At This Estimation & Approximation Question

Let’s Find The Keywords In This Question

What are the keywords in the question?

First, the number of each fruit that Chloe has to pack.

Chloe needs to pack 120 apples, 150 oranges, and 108 mangoes.

Next, she needs to pack them into identical boxes so that each type of fruit is equally distributed.

Let’s Take A Look At Part (A)

First, what do we need to solve to answer this question? We need to find the HCF of 120, 150, and 108!

There are two ways that we can find the HCF — the Prime Factorisation Method and the Ladder Method!

Finding The HCF Using The Prime Factorisation Method

Begin by writing down the three numbers — 120, 150, and 108.

Next, break them down into prime numbers.

💡 Tip When Breaking Down Numbers Into Prime Numbers 💡

If you tend to forget what prime numbers to use, always make it a habit to write down at least the first five prime numbers in a corner.
The first five prime numbers are 2, 3, 5, 7, and 11.

Take 120 and key it into your calculator.

Divide it by 2, which is the smallest prime number, and you will get 60.

Next, divide 60 by 2. It will give you 30.

Divide 30 by 2 again and you will get 15.

Now, divide 15 by 3 and you will get 5, which are prime numbers

Collect all the prime numbers and list them down.

Key in the entire expression into your calculator and check if you got 120. If not, then something must be wrong.

2 x 2 x 2 x 3 x 5 is indeed 120 so we can now write it down in index notation form.

We will get 2³ x 3 x 5.

Next, we are going to repeat the same process with 150.

Key 150 into your calculator and divide it by 2. You will get 75.

Now, divide 75 by 3 and you will get back 25.

We know that 5 x 5 is equal to 25. Now, write down all the prime numbers.

Let’s double-check: 2 x 3 x 5 x 5 = 150!

We’ve confirmed that we will get back 150 so we can now write it down in index notation.

2 x 3 x 5²

Let us repeat the process for 108.

Key in 108 into your calculator and divide it by 2. You will get 54.

Next, 54 is divisible by 2. You will get 27.

We know that 27 is divisible by 3 and you will get 9.

Finally, 9 is a product of 3 x 3.

Write down all the prime numbers.

2 x 2 x 3 x 3 x 3 = 108

Finally, write it down in index notation and you will get 2² x 3³.

Now, let us look for the HCF by looking at the prime numbers that the three given numbers have in common.

We have the 2s and 3s because 108 does not have a 5.

Check how many 2s and 3s are common among the three numbers.

2 x 3 = 6

Our HCF is 6.

Answer For Part (A)

The HCF refers to the largest number of boxes that can be packed.

Therefore, Chloe can pack 6 boxes.

Let’s Take A Look At Part (B)

Now, let us compute the total number of fruits in each box.

To do that, take the number of each type of fruit and divide it by 6.

120 apples ÷ 6 = 20 apples

150 oranges ÷ 6 = 25 oranges

108 mangoes ÷ 6 = 18 mangoes

But this is not the end, because the question is asking for the total number of fruits in each box.

20 apples + 25 oranges + 18 mangoes = 63 fruits!

Answer For Part (B)

There are 63 fruits in the boxes.

Finding The Answers Using The Ladder Method

In this question, the Ladder Method has a slight advantage because you will find the answers for (a) and (b) in one working.

Begin by drawing your ladder.

All these numbers are divisible by 2, to get 60, 75, and 54.

We cannot divide the numbers by 2 because 75 is not an even number. So, let us divide the numbers by 3. We will get 20, 25, and 18.

Finally, when you arrive at this last set of numbers, you’ll realise that they no longer have anything in common so we can now stop.

2 and 3 make up your HCF.

2 x 3 = 6

The HCF is 6 and it is also the answer for part (a) which is looking for the number of boxes that can be packed.

Now, how are we going to find the answer for part (b)? It is actually in the working. Can you spot where it is? It is in the last row!

For the ladder method, the last row will give you the number of each type of fruit, so all you have to do for part (b) is to find the total.

20 apples + 25 oranges + 18 mangoes = 63 fruits!

Conclusion

I hope that after reading this blog post, you have gained more confidence in approaching Secondary Math word problems that involve the Highest Common Factor (HCF).

As we’ve discussed, you can use either the Prime Factorisation Method or the Ladder Method.

Also, remember to read the question thoroughly to ensure that you are solving for the correct concept. Many students often get confused between HCF and LCM (Lowest Common Multiple).

Practise more of our other Estimation & Approximation questions!

Keep a lookout for more Math blog posts!