PSLE Math: H.C.F. Using the Ladder Method — Step-by-Step Guide

What is H.C.F. and Why It Matters for PSLE?

The Highest Common Factor (H.C.F.) of two or more numbers is the largest number that divides all of them exactly (with no remainder). It is sometimes called the Greatest Common Factor (G.C.F.) or Greatest Common Divisor (G.C.D.).

In the PSLE Math syllabus, H.C.F. questions appear in two main forms:

1. ▸Direct computation: "Find the H.C.F. of 48 and 72." — Straightforward; you use the Ladder Method and give a number as your answer.
2. ▸Word problems: "Mrs Tan has 48 red beads and 72 blue beads. She wants to divide them into the greatest possible number of identical groups with no beads left over. How many groups can she make?" — The H.C.F. gives the answer.

PSLE Insight: When a word problem involves dividing things into equal groups, cutting into equal pieces, or packing into identical containers with no leftovers — and asks for the greatest or maximum number — that is almost always an H.C.F. problem.

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What is the Ladder Method?

The Ladder Method (also known as the Division Method) is a systematic approach to finding H.C.F. by repeatedly dividing numbers by their common prime factors — much like climbing down rungs of a ladder. At each rung, you divide by a common factor until no further common division is possible.

The H.C.F. is then the product of all the divisors you used (the numbers on the left side of the ladder).

This method is preferred in Singapore primary schools because it is:

• ▸Easy to set out clearly for showing working.
• ▸Less prone to errors than listing factors.
• ▸Scalable — works for two or three numbers.
• ▸Accepted by PSLE markers when shown correctly.

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Step-by-Step: How to Use the Ladder Method

We will use a simple example: Find the H.C.F. of 24 and 36.

Step 1: Write the numbers side by side inside the ladder

Draw an upside-down "L" shape. Write both numbers inside, separated by a space. This is the start of your ladder.

 
2436

Step 2: Find a common factor of both numbers and write it on the left

Ask: what number divides BOTH 24 and 36 exactly? Start with the smallest prime: 2. Both 24 and 36 are even, so 2 works.

2
2436

Step 3: Divide both numbers by the factor and write results below

24 ÷ 2 = 12 and 36 ÷ 2 = 18. Write these below the line.

2
2436
 
1218

Step 4: Repeat — find a common factor of the new row

12 and 18 are still both even. Divide by 2 again.

2
2436
2
1218
 
69

Step 5: Continue until the bottom row shares no common factor

6 and 9: 6 is even but 9 is odd, so 2 doesn't work. Try 3: 6 ÷ 3 = 2 and 9 ÷ 3 = 3. Now the bottom row is 2 and 3 — they share no common factor. Stop here.

2
2436
2
1218
3
69
 
23

Step 6: Multiply all the divisors on the left to get the H.C.F.

H.C.F. = 2 × 2 × 3 = 12. The H.C.F. of 24 and 36 is 12.

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Worked Examples with Full Solutions

Example A — Straightforward

Find the H.C.F. of 48 and 72.

Thinking process: Both 48 and 72 are even → try 2. Keep dividing by 2 until one becomes odd. Then try 3.

2
4872
2
2436
2
1218
3
69
 
23

H.C.F. = 2 × 2 × 2 × 3 = 24. Answer: 24

Example B — With 5 as a factor

Find the H.C.F. of 60 and 90.

Thinking process: Both end in 0 → both divisible by 2, 3, and 5. Remember to try all prime divisors: 2, 3, 5, 7…

2
6090
3
3045
5
1015
 
23

H.C.F. = 2 × 3 × 5 = 30. Answer: 30

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H.C.F. of Three Numbers (Advanced)

The Ladder Method extends naturally to three numbers. Simply add a third column to your ladder. At each rung, the divisor must divide all three numbers evenly.

Example: Find the H.C.F. of 24, 36, and 60.

2
243660
2
121830
3
6915
 
235

H.C.F. = 2 × 2 × 3 = 12.

Important: At the last row (2, 3, 5), there is no single number that divides all three of them. The H.C.F. is only the product of divisors used on the left side of the ladder.

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PSLE-Style Word Problems Explained

Word problems are where many students lose marks — not because they cannot find the H.C.F., but because they cannot recognise when to use it.

Signal Words in H.C.F. Word Problems:

• ▸"greatest number of groups"
• ▸"maximum number of bags / trays"
• ▸"cut into equal pieces (greatest length)"
• ▸"divide equally with no remainder"
• ▸"identical sets / identical packs"
• ▸"as many as possible"

Worked Word Problem

Question: Mr Lim has 56 yellow marbles and 84 green marbles. He wants to pack them into identical bags with no marbles left over. Each bag must contain the same number of yellow marbles and the same number of green marbles. What is the greatest number of bags he can pack?

Solution:

1. ▸Identify what you need: The keyword is "greatest number of bags" with "no marbles left over". This is an H.C.F. problem. Find H.C.F. of 56 and 84.
2. ▸Apply the Ladder Method:

2
5684
2
2842
7
1421
 
23

1. ▸Calculate H.C.F.: H.C.F. = 2 × 2 × 7 = 28.
2. ▸Answer the question: The H.C.F. is 28. Therefore, Mr Lim can pack a maximum of 28 bags.

(Note: Each bag will contain 56 ÷ 28 = 2 yellow marbles and 84 ÷ 28 = 3 green marbles.)

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Common Mistakes to Avoid

1. ▸Confusing H.C.F. with L.C.M.: H.C.F. is about the GREATEST common factor (grouping or cutting into equal parts). L.C.M. is about the LEAST common multiple (reoccurrence, or when things meet again).
2. ▸Stopping the ladder too early: Always check: can the bottom row still be divided by a common factor? Only stop when the numbers share no common factor other than 1.
3. ▸Dividing by a non-prime composite number: While the method can work with composite numbers if they divide all terms, it is far safer to stick to prime divisors (2, 3, 5, 7...) to prevent errors.
4. ▸Forgetting to multiply ALL divisors: Students sometimes write down only the last divisor. H.C.F. is the product of all numbers on the left.

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Practice Questions

Q1 (Straightforward): Find the H.C.F. of 18 and 30.

• ▸Hint: Both numbers are divisible by 2, then 3.
• ▸Answer: 6 (Working: 2 × 3 = 6)

Q2 (Intermediate): Find the H.C.F. of 56 and 84.

• ▸Hint: Try dividing by 2 first, then again, then by 7.
• ▸Answer: 28 (Working: 2 × 2 × 7 = 28)

Q3 (PSLE-Style Word Problem): A baker has 72 chocolate muffins and 48 blueberry muffins. He wants to arrange them into identical trays with no muffins left over. Each tray must have the same number of each type of muffin. What is the greatest number of trays he can make?

• ▸Hint: Find the H.C.F. of 72 and 48.
• ▸Answer: 24 trays (Working: H.C.F.(72, 48) = 24. Each tray has 3 chocolate and 2 blueberry muffins.)

Q4 (PSLE-Style Word Problem): Mrs Wong has a piece of ribbon that is 90 cm long and another that is 60 cm long. She wants to cut both ribbons into equal pieces of the greatest possible length, with no ribbon wasted. How long should each piece be? How many pieces will there be altogether?

• ▸Hint: Find the H.C.F. of 90 and 60 for the length. Then divide each ribbon length by the H.C.F. to count pieces.
• ▸Answer: 30 cm each; 5 pieces altogether (Working: H.C.F.(90, 60) = 30 cm. 90 ÷ 30 = 3 pieces + 60 ÷ 30 = 2 pieces = 5 pieces total.)