Unique Prime Factorisation Concept Explained

 

❓ Theorem

Fundamental Theorem of Arithmetic

Every composite number can be expressed (factorised) as a product of prime numbers, and this factorisation is unique apart from the order in which the prime factors occur.

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🖼️ Proof Image

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✍️ Short Explanation

The theorem states that every composite number has a unique prime factorisation.

Only the order of factors may change, but the prime factors themselves remain the same 💯

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🔹 Step 1 — General Form of a Composite Number

Any composite number $x$ can be written as:

$$x = p_1 \times p_2 \times p_3 \times \cdots \times p_n$$

where:

$$p_1, p_2, p_3, \ldots$$

are prime numbers.

Example:

$$9 = 3 \times 3$$

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🔹 Step 2 — Prime Factors are Written in Order

Usually prime factors are arranged in ascending order:

$$p_1 \le p_2 \le p_3 \le \cdots \le p_n$$

This makes the factorisation easier to understand and compare.

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🔹 Step 3 — Repeated Factors can be Written in Power Form

If the same prime factor repeats, powers are used.

Example:

$$2 \times 2 \times 2 = 2^3$$

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🔹 Step 4 — Example of Prime Factorisation

Take:

$$32760$$

Prime factorisation:

$$32760 = 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 7 \times 13$$

Power form:

$$32760 = 2^3 \times 3^2 \times 5 \times 7 \times 13$$

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✅ Final Statement

$$\boxed{\text{Every composite number has a unique prime factorisation}}$$

except for the order of the prime factors.

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⭐ Key Insight

• Composite numbers can always be broken into prime numbers
• Prime factorisation is unique

🧠 Memory Line:

Every composite number has its own unique prime identity