SECONDARY 1 MATHEMATICS LEARNING GUIDE · GUIDE 13
Whole numbers have structure. A number can be decomposed into prime factors, rebuilt from those factors, compared through common factors and common multiples, and recognised through square or cube patterns. These ideas are not isolated tricks: they support fraction work, algebraic simplification, indices, standard form and later number reasoning.
This guide builds prime numbers, prime factorisation, Highest Common Factor, Lowest Common Multiple, perfect squares, perfect cubes and roots from first principles. The emphasis is on understanding what each quantity means and choosing the right representation before calculating.
Return to the Secondary Mathematics Hub. This guide connects especially to Numbers, Number Lines, Approximation and Estimation and to Algebraic Expressions and Variables.
Navigate: prime numbers · prime factorisation · HCF · LCM · squares and roots · cubes and cube roots · applications · practice · answers.
1. A factor divides exactly
A factor of a whole number divides it with no remainder. For example, the positive factors of 18 are 1, 2, 3, 6, 9 and 18.
A multiple is produced by multiplying. Multiples of 6 include 6, 12, 18, 24 and so on. The same number can therefore be a factor in one relationship and a multiple in another.
Worked contrast
Is 6 a factor of 42? Yes, because 42 ÷ 6 = 7 exactly. Is 42 a factor of 6? No. Direction matters.
2. Prime numbers have exactly two positive factors
A prime number greater than 1 has exactly two positive factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17 and 19.
The number 1 is not prime because it has only one positive factor. The number 2 is prime and is the only even prime.
Composite numbers
A composite number greater than 1 has more than two positive factors. For example, 12 is composite because it has factors 1, 2, 3, 4, 6 and 12.
3. Prime factorisation rewrites a number using only primes
Every whole number greater than 1 can be expressed as a product of primes, uniquely apart from the order of the factors. This is the fundamental theorem of arithmetic.
Worked example: 84
84 = 2 × 42 = 2 × 2 × 21 = 2² × 3 × 7.
So the prime factorisation of 84 is 2² × 3 × 7.
Factor tree check
Different factor trees must end at the same prime factorisation. Beginning with 84 = 7 × 12 gives 7 × 3 × 2 × 2, which rearranges to the same result.
4. Indices compress repeated prime factors
Instead of writing 2 × 2 × 2 × 3 × 3, write 2³ × 3². The index tells how many times the base appears as a factor.
Worked example
360 = 36 × 10 = 2² × 3² × 2 × 5 = 2³ × 3² × 5.
Expanding back gives 8 × 9 × 5 = 360, which is a useful check.
5. HCF is the greatest factor shared by all the numbers
The Highest Common Factor of two or more whole numbers is the greatest positive factor that divides all of them exactly.
Worked example: 48 and 72
48 = 2⁴ × 3. 72 = 2³ × 3².
The common prime factors use the smaller exponent present in both numbers: 2³ × 3 = 24. Therefore HCF(48,72) = 24.
Why smaller exponents?
A common factor must fit inside both numbers. For the prime 2, the first number has four copies available while the second has only three. A common factor cannot require four copies because 72 does not contain that many.
6. HCF also appears in grouping problems
Suppose 48 red counters and 72 blue counters are to be divided into the greatest possible number of identical groups, with no leftovers and the same number of each colour in every group.
The number of groups must divide both 48 and 72. The greatest possible number is their HCF, 24. Each group contains 2 red and 3 blue counters.
The phrase “greatest possible number of equal groups” is a strong HCF signal.
7. LCM is the smallest positive multiple shared by all the numbers
The Lowest Common Multiple of two or more positive whole numbers is the smallest positive number that is a multiple of each.
Worked example: 12 and 18
12 = 2² × 3. 18 = 2 × 3².
For the LCM, include every required prime using the larger exponent: 2² × 3² = 36.
Why larger exponents?
A common multiple must contain enough prime factors to be divisible by every source number. It must contain two 2s to be divisible by 12 and two 3s to be divisible by 18.
8. LCM appears in repeating-event problems
One signal repeats every 8 minutes and another every 12 minutes. If they occur together now, after how long will they next occur together?
LCM(8,12) = 24, so they next coincide after 24 minutes.
The phrase “next time together” often points to an LCM structure because we are looking for a common multiple of the repeating intervals.
9. HCF and LCM solve different questions
| Question type | Likely structure |
|---|---|
| Largest equal group size or greatest number of identical groups | HCF |
| Smallest common repeat or next simultaneous event | LCM |
| Largest exact length that measures several lengths | HCF |
| Smallest total divisible by several cycle sizes | LCM |
Do not decide from a keyword alone. Ask whether the answer must divide the given numbers or be divisible by them.
10. A square number is a number multiplied by itself
For a whole number n, n² = n × n. The first positive square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.
Square roots reverse squaring
The principal square root √a is the non-negative number whose square is a. Thus √81 = 9.
Be careful: the equation x² = 81 has two real solutions, x = 9 and x = −9, while the symbol √81 itself denotes the principal non-negative root 9.
11. Prime exponents reveal perfect squares
A positive whole number is a perfect square exactly when every exponent in its prime factorisation is even.
Worked example
144 = 2⁴ × 3². Both exponents are even, so 144 is a perfect square. √144 = 2² × 3 = 12.
Non-example
72 = 2³ × 3². The exponent 3 on 2 is odd, so 72 is not a perfect square.
12. Make a number into a square by pairing prime factors
To make 72 = 2³ × 3² into a perfect square by multiplication, the exponent of 2 must become even. Multiply by one more factor of 2.
72 × 2 = 144, a perfect square. The smallest positive whole-number multiplier is 2.
Division version
To make 72 into a perfect square by dividing by a whole-number factor, divide by 2: 72 ÷ 2 = 36.
13. Cube numbers use three equal factors
n³ = n × n × n. The first positive cubes are 1, 8, 27, 64, 125, 216, 343, 512 and 729.
Cube roots reverse cubing
∛125 = 5 because 5³ = 125. Unlike square roots, real cube roots can be negative: ∛(−125) = −5.
14. Perfect cubes have prime exponents divisible by three
A positive whole number is a perfect cube when every exponent in its prime factorisation is a multiple of 3.
Worked example
1728 = 2⁶ × 3³. Since 6 and 3 are divisible by 3, 1728 is a perfect cube. ∛1728 = 2² × 3 = 12.
Make a cube
108 = 2² × 3³. To make it a perfect cube by multiplication, the exponent of 2 must rise from 2 to 3. Multiply by 2, giving 216 = 6³.
15. Prime factorisation can simplify roots
When the relevant course has introduced simplification of surds, prime factorisation helps separate square factors.
For example, √72 = √(36×2) = 6√2.
Treat this as an extension if surds have not yet appeared in your Secondary 1 course. The underlying factorisation remains useful regardless.
16. HCF and fractions are connected
To simplify 84/126, divide numerator and denominator by their HCF.
84 = 2²×3×7 and 126 = 2×3²×7, so HCF = 2×3×7 = 42.
84/126 = 2/3.
This explains why the greatest common divisor produces the fully simplified fraction in one step.
17. LCM and common denominators are connected
To add 5/12 + 7/18, a convenient common denominator is LCM(12,18) = 36.
5/12 = 15/36 and 7/18 = 14/36, so the sum is 29/36.
Any common multiple could work as a denominator, but the LCM keeps the numbers smaller.
18. Common number-structure errors
| Error | Likely issue | Repair prompt |
|---|---|---|
| Calls 1 a prime number | Prime definition incomplete | How many positive factors must a prime have? |
| Uses largest exponents for HCF | HCF and LCM roles reversed | Must the answer fit inside both numbers? |
| Uses smallest exponents for LCM | Divisibility requirement ignored | Does the result contain enough factors for both? |
| Writes √49 = ±7 | Root symbol confused with equation solution | What does the principal square root symbol denote? |
| Calls 72 a square because it is even | Square pattern not understood | Are all prime exponents even? |
19. Practice laboratory
- List all positive factors of 24.
- State whether 29 is prime.
- Write 180 as a product of prime factors.
- Find HCF(84,126).
- Find LCM(24,36).
- Two lights flash every 18 s and 30 s. If they flash together now, when will they next flash together?
- Find √196.
- Find ∛343.
- State whether 540 is a perfect square using prime factorisation.
- Find the smallest positive whole number by which 75 must be multiplied to become a perfect square.
- Find the smallest positive whole number by which 72 must be multiplied to become a perfect cube.
- Simplify 96/144 using the HCF.
- Find a least common denominator for 7/20 and 11/30.
- A teacher has 60 red cards and 84 blue cards. What is the greatest possible number of identical groups with no leftovers?
- Explain the difference between √64 and solving x² = 64.
20. Explained answers
1. 1,2,3,4,6,8,12,24.
2. Yes. Its only positive factors are 1 and 29.
3. 180 = 2² × 3² × 5.
4. 84 = 2²×3×7, 126 = 2×3²×7. HCF = 42.
5. 24 = 2³×3, 36 = 2²×3². LCM = 72.
6. LCM(18,30) = 90 s.
7. 14.
8. 7.
9. 540 = 2²×3³×5. Not all exponents are even, so it is not a perfect square.
10. 75 = 3×5². Multiply by 3 to obtain 225 = 15².
11. 72 = 2³×3². Multiply by 3 to obtain 216 = 6³.
12. HCF(96,144) = 48, so 96/144 = 2/3.
13. LCM(20,30) = 60.
14. HCF(60,84) = 12 groups.
15. √64 denotes the principal square root 8. The equation x² = 64 has two real solutions, x = ±8.
21. Complete mixed problem
Problem: Three repeating maintenance checks occur every 18, 24 and 30 days. They occur together today. After how many days will they next coincide? Then find the greatest number of identical packs that can be made from 180 labels and 252 tags with no leftovers.
For the repeating schedule, find LCM(18,24,30). Prime factorisations are 18 = 2×3², 24 = 2³×3 and 30 = 2×3×5. The LCM is 2³×3²×5 = 360 days.
For the packs, find HCF(180,252). 180 = 2²×3²×5 and 252 = 2²×3²×7. HCF = 2²×3² = 36 packs. Each pack has 5 labels and 7 tags.
The same prime-factor representation supports two different questions. The relationship being asked determines whether smaller or larger exponents are selected.
22. Teaching the difference between HCF and LCM
Use one pair of numbers for two contrasting problems. “Make the greatest number of identical groups” and “Find the next time two cycles meet” can both use 12 and 18, but one needs HCF and the other LCM.
Ask the learner to predict whether the answer should be smaller than, between or larger than the source numbers before calculating. HCF cannot exceed the smaller positive input. LCM cannot be smaller than the larger positive input.
Return after a delay
Later, give only the story problem and remove the topic label. The learner should identify whether the unknown must divide the given numbers or be divisible by them.
23. Questions students often ask
Is every odd number prime?
No. 9, 15 and 21 are odd composite numbers.
Can HCF equal one?
Yes. Numbers with no common prime factor are coprime, such as 8 and 15.
Can LCM equal one of the numbers?
Yes. If one number is already a multiple of the other, such as 6 and 18, the LCM is 18.
Why is 0 not used as the LCM?
Although 0 is a multiple of every positive whole number, LCM is defined as the least positive common multiple.
What is the fastest check?
For HCF, verify it divides every number. For LCM, verify every source number divides the result. For squares and cubes, inspect prime exponents.
24. Return path
Prime factorisation is a foundation that keeps reappearing. It supports fractions, roots, indices and later algebra. Revisit Numbers, Number Lines, Approximation and Estimation for wider number sense and Algebraic Formulae, Substitution and Rearrangement when numerical structure begins to interact with symbolic structure.
Sources and learning boundaries
Official curriculum reference: MOE Secondary Syllabus Directory. The precise sequence and depth of prime factorisation, HCF, LCM, squares, cubes and roots differ by subject level and school.
The examples and practice questions above are independently written. Extension material such as simplifying surds should be used only when it matches the learner’s present course.
Editorial approach: Wintour House V1.0 · CivDJ · eduKate Publishing. Decompose the number, identify the relationship, select the invariant factors, verify divisibility and return the result to the question.
