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.

(N/A) Numbers are of two types: prime and composite. Prime numbers have only two factors,$1$ and the number itself,whereas composite numbers have more than two factors.
For the first expression:
$7 \times 11 \times 13+13 = 13 \times (7 \times 11 + 1) = 13 \times (77 + 1) = 13 \times 78 = 13 \times 13 \times 6$.
Since this expression has factors other than $1$ and itself (specifically $6, 13, 78$),it is a composite number.
For the second expression:
$7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1+5 = 5 \times (7 \times 6 \times 4 \times 3 \times 2 \times 1 + 1) = 5 \times (1008 + 1) = 5 \times 1009$.
Since $1009$ is a prime number,the expression has factors $5$ and $1009$ (besides $1$ and the number itself). Therefore,it is a composite number.