Question

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

Answer 1

Numbers are of two types - prime and composite. Prime numbers can be divided by $1$ and only itself, whereas composite numbers have factors other than $1$ and itself.
It can be observed that
$7 × 11 × 13 + 13 = 13 × (7 × 11 + 1)$
$= 13 × (77 + 1) = 13 × 78$
$= 13 ×13 × 6$
The given expression has $6$ and $13$ as its factors. Therefore, it is a composite number.
$7 × 6 × 5 × 4 × 3 × 2 × 1 + 5$
$= 5 ×(7 × 6 × 4 × 3 × 2 × 1 + 1)$
$= 5 × (1008 + 1)$
$= 5 ×1009$
Since, $1009$ cannot be factorised further.

Therefore, the given expression has $5$ and $1009$ as its factors. Hence, it is a composite number.