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

Numbers are of two types – composite and prime. 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.

∴ , it is a composite factor.

7×6×5×4×3×2×1 + 5 = 5×(7×6×4×3×2×1 + 1)

= 5×(1008 + 1) = 5×1009

1009 cannot be factorized further. Therefore, the given expression has 5 and 1009 as its factors. Hence, it is a composite number.