Question 6

Composite numbers have factors other than 1 and the number itself.

Step 1 — First expression analysis

Let's look at the first expression. We see that 13 is a common factor. We can take 13 out.

$7 \times 11 \times 13 + 13$

$= 13 \times (7 \times 11 + 1)$

$= 13 \times (77 + 1)$

$= 13 \times 78$

We can break down 78 further.

$= 13 \times (13 \times 6)$

$\boxed{= 13 \times 13 \times 6}$

The number has factors like 6 and 13. These are not 1 or the number itself. So, it is a composite number.

Step 2 — Second expression analysis

Now let's check the second expression. We see that 5 is a common factor. We can take 5 out.

$7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$

$= 5 \times (7 \times 6 \times 4 \times 3 \times 2 \times 1 + 1)$

Let's calculate the product inside.

$7 \times 6 \times 4 \times 3 \times 2 \times 1 = 1008$

Now we substitute this value.

$= 5 \times (1008 + 1)$

$\boxed{= 5 \times 1009}$

The number 1009 is a prime number. The number has factors like 5 and 1009. These are not 1 or the number itself. So, it is a composite number.

Answer

(i) $7 \times 11 \times 13 + 13$ is a composite number. (ii) $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$ is a composite number.