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

It can be observed that,

7 × 11 × 13 + 13 = 13 (7 × 11 + 1)

= 13(77 + 1)

= 13 × 78

= 13 × 13 × 6 × 1

= 13 × 13 × 2 × 3 × 1

The given number has 2, 3, 13, and 1 as its factors.

For the number to be prime, it should have only two factors — 1 and the number itself.

As the given number has more that two factors,

∴ 7 × 11 × 13 + 13 is a composite number.

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

= 5 × (1008 + 1)

= 5 × 1009 × 1

1009 cannot be factorized further.

The given number has 1, 5, and 1009 as its factors.

For the number to be prime, it should have only two factors — 1 and the number itself.

As the given number has more that two factors,

∴ 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 is a composite number.

Hence, proved that 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.