Explain why \(7 × 11 × 13 + 13\) and \(7 × 6 × 5 × 4 × 3 × 2 × 1 + 5\) are composite numbers.

There are two types of number prime and composite
Prime numbers can be divided by 1 and only itself, whereas composite numbers have factors other than 1 and itself.
\(\therefore \) \(7 × 11 × 13 + 13\)
= \(13 × (7 × 11 + 1)\)
= \(13 × (77 + 1)\)
= \(13 × 78 = 13 × 13 × 6\)
6 and 13 are the factors of \(7 × 11 × 13 + 13\)

And we know that numbers which have factors other than one and itself are composite number
\(7 × 6 × 5 × 4 × 3 × 2 × 1 + 5\)
=\(5 × (7 × 6 × 4 × 3 × 2 × 1 + 1)\)
= \(5 × (1008 + 1)\)
= \(5 × 1009 \)

1009 is a prime number and cannot be factorized further

So, 5 and 1009 are the factors of \(7 × 6 × 5 × 4 × 3 × 2 × 1 + 5\)

And we know that numbers which have factors other than one and itself are composite number