Scholars & Parents
giftedGrades 3–5

Number Theory — Gifted & Enrichment

What Is This?

Number theory is the study of the properties and relationships of whole numbers. It is one of the oldest and deepest branches of mathematics — and it is full of patterns, puzzles, and surprising results that mathematicians are still exploring today. At the heart of number theory are a few powerful ideas: factors, multiples, prime numbers, divisibility, greatest common factor (GCF), and least common multiple (LCM). Mastering these ideas gives you tools that reach across all of mathematics.

Factors and Multiples

Factors

The factors of a number are all the whole numbers that divide into it exactly, leaving no remainder.

Example: Factors of 36: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 — nine factors in total.

Key insight: Factors always come in pairs (except when the number is a perfect square, in which case the square root pairs with itself). This is why 36 has an odd number of factors — 6 pairs with itself.

Factor pairs of 36: (1, 36), (2, 18), (3, 12), (4, 9), (6, 6)

Multiples

The multiples of a number are what you get when you multiply it by 1, 2, 3, 4, ...

Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, ...

Multiples go on forever. Factors are always finite.

Prime and Composite Numbers

A prime number has exactly two factors: 1 and itself.

A composite number has more than two factors — it can be divided by numbers other than 1 and itself.

1 is neither prime nor composite — this is an important exception.

The First 20 Prime Numbers

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71

Notice: 2 is the only even prime number. Every other even number is divisible by 2 and is therefore composite.

Why Is 1 Not Prime?

The Fundamental Theorem of Arithmetic states that every whole number greater than 1 can be written as a unique product of prime numbers. If 1 were prime, this uniqueness would break down (you could keep multiplying by 1 without changing the value). Excluding 1 from primes keeps the theorem clean and universally true.

Prime Factorisation

Every composite number can be expressed as a product of prime factors. This is called its prime factorisation.

Method — Factor Trees:

To find the prime factorisation of 360:

        360
       /    \
      4     90
     / \   /  \
    2   2 9    10
         / \ /  \
        3  3 2   5

Prime factorisation of 360: 2³ × 3² × 5

Check: 8 × 9 × 5 = 360 ✓

Key property: The prime factorisation of a number is unique — there is only one way to write it as a product of primes (the order doesn't matter, but the factors are fixed). This is the Fundamental Theorem of Arithmetic.

Divisibility Rules

Divisibility rules let you quickly test whether a number divides evenly into another — without doing long division.

Divisor Rule Example
2 Last digit is even 738 ÷ 2 ✓ (8 is even)
3 Sum of digits divisible by 3 471: 4+7+1=12, 12÷3 ✓
4 Last two digits divisible by 4 1,324: 24÷4=6 ✓
5 Last digit is 0 or 5 735 ÷ 5 ✓
6 Divisible by both 2 and 3 438: even ✓, 4+3+8=15 ✓
7 Double the last digit, subtract from rest; repeat 161: 16 − (1×2) = 14, 14÷7 ✓
8 Last three digits divisible by 8 4,512: 512÷8=64 ✓
9 Sum of digits divisible by 9 729: 7+2+9=18, 18÷9 ✓
10 Last digit is 0 4,570 ÷ 10 ✓
11 Alternating sum of digits divisible by 11 2,816: 2−8+1−6=−11 ✓

The rule for 7 deserves special mention. Take 343: 34 − (3×2) = 34 − 6 = 28. 28 ÷ 7 = 4. So 343 is divisible by 7. (343 = 7³, so this is correct.)

Greatest Common Factor (GCF)

The GCF (also called HCF — Highest Common Factor) of two or more numbers is the largest number that divides into all of them exactly.

Method 1 — List Factors

GCF of 24 and 36:

  • Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
  • Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
  • Common factors: 1, 2, 3, 4, 6, 12
  • GCF = 12

Method 2 — Prime Factorisation

GCF of 180 and 252:

  • 180 = 2² × 3² × 5
  • 252 = 2² × 3² × 7
  • GCF = 2² × 3² = 4 × 9 = 36

Take the lowest power of each prime that appears in BOTH factorisations.

Method 3 — Euclidean Algorithm (Most Efficient)

GCF(252, 180):

  • 252 = 1 × 180 + 72
  • 180 = 2 × 72 + 36
  • 72 = 2 × 36 + 0

When the remainder is 0, the last non-zero remainder is the GCF. GCF = 36.

The Euclidean Algorithm is the fastest method for large numbers and is one of the oldest algorithms in mathematics (over 2,300 years old).

Least Common Multiple (LCM)

The LCM of two or more numbers is the smallest number that is a multiple of all of them.

Method 1 — List Multiples

LCM of 8 and 12:

  • Multiples of 8: 8, 16, 24, 32, 40...
  • Multiples of 12: 12, 24, 36...
  • LCM = 24

Method 2 — Prime Factorisation

LCM of 180 and 252:

  • 180 = 2² × 3² × 5
  • 252 = 2² × 3² × 7
  • LCM = 2² × 3² × 5 × 7 = 4 × 9 × 5 × 7 = 1,260

Take the highest power of each prime that appears in EITHER factorisation.

The GCF–LCM Relationship

For any two numbers a and b:

GCF(a, b) × LCM(a, b) = a × b

Check: GCF(180, 252) × LCM(180, 252) = 36 × 1,260 = 45,360 = 180 × 252 ✓

This relationship is powerful — if you know the GCF, you can immediately find the LCM without a separate calculation.

Perfect Numbers, Abundant, and Deficient

A number is perfect if the sum of its proper factors (all factors except itself) equals the number.

  • 6: proper factors 1+2+3 = 6 ✓ (perfect)
  • 28: proper factors 1+2+4+7+14 = 28 ✓ (perfect)

A number is abundant if the sum of its proper factors exceeds it: 12 → 1+2+3+4+6 = 16 > 12.

A number is deficient if the sum of its proper factors is less than it: 8 → 1+2+4 = 7 < 8.

Most numbers are deficient. Perfect numbers are extraordinarily rare — only 51 are known, and every known perfect number is even.

The Sieve of Eratosthenes

The Sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any limit.

Steps to find all primes up to 100:

  1. Write all numbers from 2 to 100
  2. Circle 2 (prime). Cross out all multiples of 2 (4, 6, 8...)
  3. Circle 3 (prime). Cross out all multiples of 3 not already crossed
  4. Circle 5. Cross out all multiples of 5
  5. Circle 7. Cross out all multiples of 7
  6. All remaining uncrossed numbers are prime

Key insight: You only need to check primes up to √100 = 10. Any composite number ≤ 100 must have a prime factor ≤ 10. So checking primes 2, 3, 5, and 7 is sufficient.

There are 25 prime numbers between 1 and 100.

Competition Corner

The Number of Factors Formula

If n = p₁^a × p₂^b × p₃^c × ..., then the number of factors of n is: (a+1)(b+1)(c+1)...

Example: How many factors does 360 have? 360 = 2³ × 3² × 5¹ Number of factors = (3+1)(2+1)(1+1) = 4 × 3 × 2 = 24

This is far faster than listing factors for large numbers.

Finding Perfect Squares from Factor Count

A number has an odd number of factors if and only if it is a perfect square. This is because only perfect squares have a factor that pairs with itself.

Consecutive Integers and GCF

The GCF of any two consecutive integers is always 1. Consecutive integers share no common factors except 1 — they are called coprime or relatively prime.

This means fractions with consecutive integer numerators and denominators are always in lowest terms: 99/100 cannot be simplified.

Key Takeaways

  • Every whole number > 1 is either prime or composite — never both, and 1 is neither
  • Prime factorisation is unique (Fundamental Theorem of Arithmetic)
  • GCF: use the lowest power of shared primes; LCM: use the highest power of all primes
  • GCF × LCM = product of the two numbers — use this relationship to save time
  • Divisibility rules let you test factors without division
  • The Euclidean Algorithm is the most efficient way to find GCF for large numbers
  • A number's factor count formula: if n = p^a × q^b, factors = (a+1)(b+1)
  • Perfect squares have an odd number of factors
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