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.
Step-by-Step Solution
Key Concept: A number is composite if it has factors other than 1 and itself (Fundamental Theorem of Arithmetic).
Stepwise Solution:
$7 \times 11 \times 13 + 13 = 13 \times (7 \times 11 + 1) = 13 \times (77 + 1) = 13 \times 78$. Since it has factors $13$ and $78$ other than $1$ and itself, it is composite. [1.0 Mark]
$7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5 = 5 \times (7 \times 6 \times 4 \times 3 \times 2 \times 1 + 1) = 5 \times (1008 + 1) = 5 \times 1009$. Since it has $5$ as a factor other than $1$ and itself, it is composite. [1.0 Mark]
Marking Scheme:
• Factoring out 13 and explaining first number composite: 1.0 Mark
• Factoring out 5 and explaining second number composite: 1.0 Mark
Correct Answer: