Number structure is the hidden architecture under a great deal of later Mathematics. Prime factorisation explains divisibility. HCF and LCM organise repeated cycles. Roots connect multiplication back to length and scale. Rational and irrational numbers explain what kind of number an answer actually is.
This Secondary 3 Mathematics Learning Guide develops prime factorisation, HCF, LCM, squares, cubes, square roots, cube roots, rational numbers, irrational numbers and ordering on the real number line. It belongs to the Secondary Mathematics Hub.
The current 2027 SEC G3 Mathematics syllabus listing from SEAB identifies Mathematics as K310. Its Number and Algebra content includes primes and prime factorisation, HCF and LCM, roots, rational and real numbers, number-line ordering, approximation, standard form and indices. This article concentrates on the structural number ideas not already covered deeply in the earlier indices guide.
Prime Numbers Are Building Blocks
A prime number is a positive integer greater than 1 with exactly two positive factors: 1 and itself. Numbers such as 2, 3, 5, 7 and 11 are prime. The number 1 is not prime because it has only one positive factor.
Every positive integer greater than 1 can be expressed as a product of primes. Apart from the order of the factors, that prime factorisation is unique.
Worked Example 1: Prime Factorisation
Express 756 as a product of prime factors. Divide systematically: 756=2×378=2²×189=2²×3×63=2²×3³×7.
Therefore 756=2²×3³×7. Multiplying the factors back gives 4×27×7=756, providing a direct check.
HCF Uses the Common Prime Factors at Their Smallest Powers
The highest common factor is the greatest positive integer dividing all the given numbers exactly. Once numbers are written in prime-factor form, keep only primes appearing in every number and use the smallest exponent among them.
For 72=2³×3² and 120=2³×3×5, the HCF is 2³×3=24.
LCM Uses Every Required Prime at Its Largest Power
The lowest common multiple is the smallest positive integer that is a multiple of all the given numbers. In prime-factor form, include every prime appearing in any number and use the largest exponent required.
For the same 72=2³×3² and 120=2³×3×5, the LCM is 2³×3²×5=360.
Worked Example 2: HCF and LCM Together
Find the HCF and LCM of 84 and 126. Factorise: 84=2²×3×7 and 126=2×3²×7.
HCF=2×3×7=42. LCM=2²×3²×7=252.
A useful check for two positive integers is HCF×LCM = product of the two numbers. Here 42×252=10,584 and 84×126=10,584.
HCF Solves Largest-Equal-Group Problems
When two or more quantities must be divided into the largest possible equal groups with no remainder, HCF is usually the relevant structure.
Example: 84 red beads and 126 blue beads are packed into identical sets using all beads. The greatest number of identical sets is HCF(84,126)=42. Each set contains 2 red and 3 blue beads.
LCM Solves Repeating-Cycle Problems
When two repeating events must occur together again, LCM is usually the relevant structure.
Example: One signal flashes every 18 seconds and another every 30 seconds. LCM(18,30)=90. If they flash together now, they will next flash together in 90 seconds.
Perfect Squares and Square Roots
A perfect square has even exponents in its prime factorisation. For example, 3600=2⁴×3²×5², so √3600=2²×3×5=60.
If a prime exponent is odd, the number is not a perfect square. This makes prime factorisation a structural test rather than a calculator guess.
Worked Example 3: Smallest Multiplier for a Perfect Square
Find the smallest positive integer k such that 180k is a perfect square. Since 180=2²×3²×5, only the exponent of 5 is odd.
Multiply by 5 to obtain 2²×3²×5². Therefore k=5.
Perfect Cubes and Cube Roots
A perfect cube has prime exponents that are multiples of 3. For example, 1728=2⁶×3³, so ∛1728=2²×3=12.
This provides the parallel structure: square roots divide even exponents by 2; cube roots divide exponents that are multiples of 3 by 3.
Worked Example 4: Smallest Multiplier for a Perfect Cube
Find the smallest positive integer m such that 54m is a perfect cube. 54=2×3³.
The exponent of 2 is 1 and must become 3. Multiply by 2²=4. Hence m=4 and 54×4=216=6³.
Rational Numbers
A rational number can be written as p/q where p and q are integers and q≠0. Integers, terminating decimals and recurring decimals are rational.
Examples include −4, 7/9, 0.125 and 0.333… . Every integer is rational because n=n/1.
Irrational Numbers
An irrational number cannot be written as a ratio of two integers. Its decimal expansion does not terminate and does not repeat in a fixed cycle.
Examples include √2, √3 and π. A square root is not automatically irrational: √49=7 is rational. The key question is whether the root simplifies to a rational value.
Real Numbers Combine Rational and Irrational Numbers
The real number line contains both rational and irrational numbers. Every point on the ordinary continuous number line corresponds to a real number.
This is why values such as √2 and π can be located approximately on a number line even though their decimal expansions never terminate.
Worked Example 5: Classify Numbers
Classify −8, 0.4, √81 and √10.
−8 is an integer and rational. 0.4=2/5 is rational. √81=9 is an integer and rational. √10 is irrational. All four are real numbers.
Ordering Exact and Approximate Values
Sometimes exact numbers must be compared using approximations. For example, √10≈3.162, π≈3.142 and 22/7≈3.143. Therefore π<22/7<√10.
Use enough precision to distinguish the values, but do not replace an exact form permanently unless approximation is requested.
Negative Numbers and Order
On the number line, a number farther right is greater. Thus −3>−8 even though 8 has the larger absolute value.
Absolute value measures distance from zero, not signed order. |−8|=8 while |−3|=3, but −8<−3.
Common Errors
Treating 1 as prime: repair by using the definition of exactly two positive factors.
Using largest powers for HCF: repair by remembering that the HCF must divide every number, so it can use only the smallest common exponents.
Assuming every square root is irrational: repair by checking whether the radicand is a perfect square.
Comparing negative numbers by absolute size: repair by returning to the number line.
Independent Practice
1. Express 540 as a product of prime factors.
2. Find HCF(96,144).
3. Find LCM(72,90).
4. Find the smallest k such that 150k is a perfect square.
5. Find the smallest m such that 40m is a perfect cube.
6. Classify √64, √7 and 0.121212… as rational or irrational.
7. Put π, 3.14 and √10 in ascending order.
8. Two alarms sound every 24 min and 36 min. If together now, when next together?
9. 90 apples and 150 oranges are packed into the greatest number of identical bags. How many bags and how many of each fruit per bag?
10. Explain why every integer is rational.
Explained Answers
1. 540=2²×3³×5.
2. 96=2⁵×3, 144=2⁴×3², so HCF=2⁴×3=48.
3. 72=2³×3², 90=2×3²×5, so LCM=2³×3²×5=360.
4. 150=2×3×5², so multiply by 2×3=6.
5. 40=2³×5, so multiply by 5²=25.
6. √64=8 rational; √7 irrational; recurring decimal 0.121212… rational.
7. 3.14<π<√10.
8. LCM(24,36)=72 min.
9. HCF(90,150)=30 bags; 3 apples and 5 oranges per bag.
10. Any integer n can be written as n/1.
Continue the Secondary 3 Learning Route
Continue with Algebraic Patterns, nth-Term Rules and Identities, Simultaneous Linear Equations and Modelling, and Compound Interest, Repeated Growth and Financial Reasoning.
Number structure becomes useful when factorisation explains why a method works rather than merely producing an answer. Return to the Secondary Mathematics Hub.
