Introduction

• Every whole number greater than $1$ is built from smaller, indivisible pieces.
• These pieces are prime numbers, and they act like the “atoms” of arithmetic.
• The Fundamental Theorem of Arithmetic (FTA) tells us that every integer has a unique prime factorization.

What Are Prime Numbers?

• A prime number is a whole number greater than $1$ that has exactly two positive divisors:
  • $1$
  • itself
• Examples:
  • $2, 3, 5, 7, 11, 13, 17, 19, 23$
• Non‑examples:
  • $1$ (only one divisor)
  • $4 = 2 \times 2$
  • $15 = 3 \times 5$
• Key idea: primes cannot be broken down into smaller whole-number factors.

Composite Numbers and the Need for Building Blocks

• A composite number is a whole number greater than $1$ that is not prime.
• It can be written as a product of smaller whole numbers.
• Examples:
  • $6 = 2 \times 3$
  • $12 = 2 \times 2 \times 3$
  • $100 = 2 \times 2 \times 5 \times 5$
• Why this matters:
  • Composite numbers are “built” from primes.
  • Understanding primes helps us understand the structure of all integers.

Historical Perspectives on Prime Numbers

• Ancient Greeks (Euclid, c. 300 BCE):
  • Proved there are infinitely many primes.
  • Studied factorization and divisibility.
• Middle Ages:
  • Primes used in early number theory and algebra.
• Modern era:
  • Primes became central to cryptography and computer science.
• Across history, primes have been viewed as mysterious, fundamental, and essential.

The Idea of “Atoms” in Mathematics

• Atoms in chemistry:
  • Smallest building blocks of matter.
• Primes in arithmetic:
  • Smallest building blocks of integers.
• Similarities:
  • Both combine to form more complex structures.
  • Both are “irreducible.”
• This analogy helps beginners see why primes are so important.

Statement of the Fundamental Theorem of Arithmetic

The theorem says:

• Every integer greater than $1$ can be written as a product of prime numbers, and this factorization is unique except for the order of the factors.

In symbols:

If $$n = p_1 p_2 \cdots p_k,$$ then the primes $p_i$ are uniquely determined up to rearrangement.

Why Uniqueness Matters

• Without uniqueness, arithmetic would fall apart.
• Imagine if:
  • $12$ could be factored in two completely different ways using different primes.
• Consequences of uniqueness:
  • Reliable simplification of fractions.
  • Consistent algebraic manipulation.
  • Predictable number patterns.
• Uniqueness is the backbone of number theory.

How Factorization Works: Step‑by‑Step Examples

Example 1: Factor $60$

• Divide by the smallest prime:
  • $60 = 2 \times 30$
  • $30 = 2 \times 15$
  • $15 = 3 \times 5$
• Final factorization:
  • $$60 = 2^2 \times 3 \times 5$$

Example 2: Factor $84$

• $84 = 2 \times 42$
• $42 = 2 \times 21$
• $21 = 3 \times 7$
• Final:
  • $$84 = 2^2 \times 3 \times 7$$

Example 3: Factor $210$

• $210 = 2 \times 105$
• $105 = 3 \times 35$
• $35 = 5 \times 7$
• Final:
  • $$210 = 2 \times 3 \times 5 \times 7$$

Common Misconceptions About Prime Factorization

• “1 is prime.”
  • No — including $1$ would break uniqueness.
• “Prime factorizations can differ depending on how you start.”
  • The process may differ, but the final primes are always the same.
• “Large numbers must have large prime factors.”
  • Not necessarily:
    • $1001 = 7 \times 11 \times 13$
• “Prime factorization is only for math class.”
  • It’s used in many real-world systems.

Applications of Prime Factorization in Everyday Mathematics

• Simplifying fractions:
  • $$\frac{18}{24} = \frac{2 \times 3^2}{2^3 \times 3} = \frac{3}{4}$$
• Finding greatest common divisors (GCDs).
• Finding least common multiples (LCMs).
• Understanding repeating decimals.
• Checking divisibility patterns.

Prime Factorization in Modern Technology (Cryptography, Coding, etc.)

• RSA encryption:
  • Relies on the difficulty of factoring very large numbers.
  • Public key uses a product of two large primes.
• Hashing and checksums:
  • Use modular arithmetic built on prime properties.
• Error‑correcting codes:
  • Use prime-based structures to detect and fix data errors.
• Random number generation:
  • Often uses prime moduli for better distribution.

Calculator

Exercises

1. Factor each number into primes:
   • (a) $48$
   • (b) $72$
   • (c) $150$

   Solution

   Factor each number into primes

   (a) $48$

   • $48 = 2 \times 24$
   • $24 = 2 \times 12$
   • $12 = 2 \times 6$
   • $6 = 2 \times 3$
   • So $$48 = 2^4 \times 3$$

   (b) $72$

   • $72 = 2 \times 36$
   • $36 = 2 \times 18$
   • $18 = 2 \times 9$
   • $9 = 3 \times 3$
   • So $$72 = 2^3 \times 3^2$$

   (c) $150$

   • $150 = 10 \times 15$
   • $10 = 2 \times 5$
   • $15 = 3 \times 5$
   • So $$150 = 2 \times 3 \times 5^2$$
2. Write the prime factorization of $360$ using exponents.

   Solution

   Prime factorization of $360$

   • $360 = 36 \times 10$
   • $36 = 6 \times 6 = 2 \times 3 \times 2 \times 3$
   • $10 = 2 \times 5$
   • Collect primes:
     • $360 = 2^3 \times 3^2 \times 5$
3. True or false:
   • (a) $1$ is a prime number.
   • (b) Every even number greater than $2$ is composite.
   • (c) Prime factorizations are unique.

   Solution

   True or false
   (a) $1$ is a prime number.

   • False. A prime must have exactly two positive divisors ($1$ and itself). The number $1$ has only one divisor: itself.

   (b) Every even number greater than $2$ is composite.

   • True. Any even number greater than $2$ is divisible by $2$ and at least one other number, so it is composite.

   (c) Prime factorizations are unique.

   • True. This is exactly what the Fundamental Theorem of Arithmetic states.
4. Simplify using prime factorization: $$\frac{84}{126}$$

   Solution

   Simplify using prime factorization: $\dfrac{84}{126}$

   • Factor each number:
     • $84 = 2^2 \times 3 \times 7$
     • $126 = 2 \times 3^2 \times 7$
     • Write the fraction: $$ \frac{84}{126} = \frac{2^2 \times 3 \times 7}{2 \times 3^2 \times 7} $$
     • Cancel common factors:
       • Cancel one $2$ (leaves $2$ on top)
       • Cancel one $3$ (leaves $3$ on bottom)
       • Cancel $7$ completely
     • Result: $$ \frac{84}{126} = \frac{2}{3} $$
5. Find the GCD and LCM of $18$ and $30$ using prime factorizations.

   Solution

   GCD and LCM of $18$ and $30$

   Prime factorizations:

   • $18 = 2 \times 3^2$
   • $30 = 2 \times 3 \times 5$

   GCD (greatest common divisor):

   • Take the smallest power of each prime that appears in both:
     • $2^1$ (both have at least one $2$)
     • $3^1$ (both have at least one $3$)
   • So: $$ \gcd(18, 30) = 2 \times 3 = 6 $$

   LCM (least common multiple):

   • Take the largest power of each prime that appears in either:
     • $2^1$ (max from $18$ and $30$)
     • $3^2$ (from $18$)
     • $5^1$ (from $30$)
   • So: $$ \mathrm{lcm}(18, 30) = 2 \times 3^2 \times 5 = 90 $$
6. Factor $231$ and $385$. What do you notice about the primes involved?

   Solution

   Factor $231$ and $385$. What do you notice?

   $231$:

   • Try dividing by $3$:
   • $231 = 3 \times 77$
   • $77 = 7 \times 11$
   • So: $$ 231 = 3 \times 7 \times 11 $$

   $385$:

   • Ends with $5$, so divisible by $5$.
   • $385 = 5 \times 77$
   • $77 = 7 \times 11$
   • So: $$ 385 = 5 \times 7 \times 11 $$

   Observation:

   • Both numbers share the prime factors $7$ and $11$.
   • $231$ and $385$ differ only by a factor of $3$ vs $5$.
7. Challenge: Find a number between $200$ and $300$ that has exactly three distinct prime factors.

   Solution

   Find a number between $200$ and $300$ with exactly three distinct prime factors

   • One way is to choose three small primes and multiply them, then check if the result is between $200$ and $300$.
   • Example: $3, 5, 17$
     • $3 \times 5 \times 17 = 15 \times 17 = 255$