Number Theory and Basic Operations
Number theory is the foundation of all quantitative reasoning and arithmetic. For the Post-UTME examination, a strong grasp of number properties, divisibility rules, factors and multiples, and basic operations with integers, fractions, and decimals is essential. This topic covers the building blocks that underpin every other quantitative concept you will encounter in the examination.
Types of Numbers
Understanding the different categories of numbers is fundamental:
- Natural Numbers (N): The counting numbers: 1, 2, 3, 4, … (some definitions include 0)
- Whole Numbers: Natural numbers plus zero: 0, 1, 2, 3, …
- Integers (Z): All positive and negative whole numbers including zero: …, -3, -2, -1, 0, 1, 2, 3, …
- Rational Numbers (Q): Numbers that can be expressed as a fraction p/q where p and q are integers and q ≠ 0. All integers, terminating decimals, and repeating decimals are rational.
- Irrational Numbers: Numbers that cannot be expressed as a simple fraction. Examples include √2, π, e. Their decimal expansions are non-terminating and non-repeating.
- Real Numbers (R): The union of rational and irrational numbers.
Important Properties of Real Numbers
| Property | Addition | Multiplication |
|---|---|---|
| Commutative | a + b = b + a | a × b = b × a |
| Associative | (a + b) + c = a + (b + c) | (a × b) × c = a × (b × c) |
| Identity | a + 0 = a | a × 1 = a |
| Inverse | a + (−a) = 0 | a × (1/a) = 1 (a ≠ 0) |
| Distributive | a(b + c) = ab + ac |
Divisibility Rules
For the Post-UTME, you must know these standard divisibility rules:
- 2: Last digit is even (0, 2, 4, 6, 8)
- 3: Sum of digits is divisible by 3
- 4: Last two digits form a number divisible by 4
- 5: Last digit is 0 or 5
- 6: Divisible by both 2 and 3
- 7: Double the last digit and subtract from the rest; if result is divisible by 7
- 8: Last three digits form a number divisible by 8
- 9: Sum of digits is divisible by 9
- 10: Last digit is 0
- 11: Difference between the sum of alternate digits is a multiple of 11
Example: Is 2,466 divisible by 3? Sum of digits = 2 + 4 + 6 + 6 = 18. Since 18 is divisible by 3, 2,466 is divisible by 3.
Factors and Multiples
Definitions
- Factor (Divisor): A number that divides another exactly. If a divides b, we write a | b.
- Multiple: A number that is the product of a given number and an integer.
- Prime Number: A natural number greater than 1 with exactly two factors (1 and itself).
- Composite Number: A natural number greater than 1 that is not prime.
- Co-prime Numbers: Two numbers with GCD = 1 (e.g., 8 and 15)
Finding HCF and LCM
Methods: Prime Factorization and Division Method
Example using Prime Factorization: Find HCF and LCM of 12 and 18.
- 12 = 2² × 3
- 18 = 2 × 3²
- HCF = product of common factors with lowest powers = 2¹ × 3¹ = 6
- LCM = product of all factors with highest powers = 2² × 3² = 36
HCF × LCM = Product of the two numbers (useful for verification)
Even and Odd Numbers
- Even: Divisible by 2 (ends in 0, 2, 4, 6, 8)
- Odd: Not divisible by 2 (ends in 1, 3, 5, 7, 9)
Key Rules:
- Even + Even = Even
- Odd + Odd = Even
- Even + Odd = Odd
- Even × Even = Even
- Even × Odd = Even
- Odd × Odd = Odd
PEMDAS / Order of Operations
When evaluating expressions, follow BODMAS/PEMDAS:
- Parentheses/Brackets
- Orders/Exponents
- Multiplication and Division (left to right)
- Addition and Subtraction (left to right)
Example: Evaluate 3 + 4 × 2² − 6 ÷ 3 = 3 + 4 × 4 − 2 [Exponents first] = 3 + 16 − 2 [Multiplication and division] = 19 − 2 [Addition] = 17
Key Post-UTME Exam Facts
- BODMAS is the Nigerian examination term for order of operations
- HCF × LCM = Product of the two numbers — a frequently tested relationship
- 1 is neither prime nor composite
- The only consecutive primes are 2 and 3
- ⚡ Exam tip: Post-UTME frequently tests divisibility rules, HCF/LCM, and order of operations. Learn the rules for 7 and 11 — they are commonly tested and often confused.
🟢 Lite — Quick Review (1h–1d)
Rapid summary for last-minute revision before your exam.
- Number types: Natural → Whole → Integer → Rational → Real (each extends the previous)
- Divisibility rules: 2 (even), 3 (digit sum), 4 (last 2 digits), 5 (0/5), 6 (2+3), 8 (last 3), 9 (digit sum), 10 (ends 0), 11 (alternating digit sum)
- HCF: Product of common prime factors with lowest powers (use division method or prime factorization)
- LCM: Product of all prime factors with highest powers
- HCF × LCM = Product of the two numbers (verify answers with this!)
- Order of operations: BODMAS/PEMDAS
- ⚡ Exam tip: Always verify HCF/LCM using the product relationship
🟡 Standard — Regular Study (2d–2mo)
Standard content for students with a few days to months.
Integer Arithmetic and Negative Numbers
When working with integers, especially negative numbers, follow these rules:
Addition:
- Same sign: Add the numbers, keep the sign
- Different signs: Subtract smaller from larger, use the sign of the larger number
Multiplication/Division:
- Same sign → Positive
- Different signs → Negative
The Euclidean Algorithm for HCF
The Euclidean algorithm is a fast method for finding the HCF of large numbers:
Example: Find HCF of 825 and 735:
825 ÷ 735 = 1 remainder 90
735 ÷ 90 = 8 remainder 15
90 ÷ 15 = 6 remainder 0
HCF = 15🔴 Extended — Deep Study (3mo+)
Comprehensive coverage for students on a longer study timeline.
Number Theory: Advanced Concepts
Prime Factorization Method for HCF and LCM
For three or more numbers, the LCM is found by taking the highest power of each prime factor appearing in any number.
Example: Find LCM of 12, 15, and 20.
- 12 = 2² × 3
- 15 = 3 × 5
- 20 = 2² × 5
- LCM = 2² × 3 × 5 = 60
Fermat’s Little Theorem
If p is prime and a is not divisible by p, then a^(p-1) ≡ 1 (mod p). This is useful for divisibility testing in modular arithmetic.
Squares and Cubes to Memorize
| Number | Square | Cube | √(perfect square) |
|---|---|---|---|
| 1 | 1 | 1 | 1 |
| 2 | 4 | 8 | √2 ≈ 1.414 |
| 3 | 9 | 27 | √3 ≈ 1.732 |
| 4 | 16 | 64 | 2 |
| 5 | 25 | 125 | √5 ≈ 2.236 |
| 6 | 36 | 216 | √6 ≈ 2.449 |
| 7 | 49 | 343 | √7 ≈ 2.646 |
| 8 | 64 | 512 | 2√2 ≈ 2.828 |
| 9 | 81 | 729 | 3 |
| 10 | 100 | 1000 | √10 ≈ 3.162 |
| 11 | 121 | 1331 | |
| 12 | 144 | 1728 |
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