Composite Numbers Explained Using Factorisation

 

❓ Question

Explain why:

$$7 \times 11 \times 13 + 13$$

and

$$7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$$

are composite numbers.

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🖼️ Solution Image

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✍️ Short Solution

A number is called composite if it has more than two factors.

We will factorise both expressions to show that they have common factors 💯

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🔹 Step 1 — First Number

$$7 \times 11 \times 13 + 13$$

Take common factor $13$:

$$= 13(7 \times 11 + 1)$$
$$= 13(77 + 1)$$
$$= 13 \times 78$$

👉 Since it can be written as a product of two integers greater than 1, it is a composite number.

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🔹 Step 2 — Second Number

$$7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$$

Take common factor $5$:

$$= 5(7 \times 6 \times 4 \times 3 \times 2 \times 1 + 1)$$ $$= 5(1008 + 1)$$ $$= 5 \times 1009$$

👉 Again, it is expressed as a product of two integers greater than 1.

Hence, it is also composite.

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✅ Final Answer

$$7 \times 11 \times 13 + 13 = 13 \times 78$$ $$7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5 = 5 \times 1009$$

Therefore, both numbers are composite numbers.

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⭐ Key Insight

If a number can be factorised into two integers greater than 1, then it is composite.

🧠 Memory Line:

More than two factors ⇒ Composite number