Question 6

Observe the expression $7 \times 11 \times 13 + 13$. Notice that the term $13$ appears in both parts of the sum. The first part is $7 \times 11 \times 13$ and the second part is simply $13$ (which can be written as $1 \times 13$).

Apply the distributive property $ab + ac = a(b+c)$ by factoring out $13$:

$$\begin{aligned}
7 \times 11 \times 13 + 13 &= 13 \times (7 \times 11) + 13 \times 1  \\[0.6em]
&= 13 \times (7 \times 11 + 1)
\end{aligned}$$

Calculate the value inside the parentheses:

$$\begin{aligned}
7 \times 11 + 1 &= 77 + 1  \\[0.6em]
&= 78
\end{aligned}$$

Thus, the original expression simplifies to:

$$13 \times 78$$

We have expressed the number as the product of two integers: $13$ and $78$. Both $13$ and $78$ are integers greater than $1$. Since the number can be written as a product of two integers other than $1$ and itself, it is by definition a composite number.

Observe the expression $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$. Notice that the term $5$ is a factor in the first part of the sum. The first part is $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ and the second part is $5$ (which can be written as $1 \times 5$).

Apply the distributive property by factoring out $5$:

$$\begin{aligned}
7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5 &= 5 \times (7 \times 6 \times 4 \times 3 \times 2 \times 1) + 5 \times 1  \\[0.6em]
&= 5 \times (7 \times 6 \times 4 \times 3 \times 2 \times 1 + 1)
\end{aligned}$$

Calculate the product inside the parentheses. Note that $7 \times 6 \times 4 \times 3 \times 2 \times 1$ is equal to $5040$ (which is $7!$ divided by $5$). Let's compute it step-by-step or simply recognize it as an integer $K$.

$$\begin{aligned}
7 \times 6 \times 4 \times 3 \times 2 \times 1 &= 5040  \\[0.6em]
5040 + 1 &= 5041
\end{aligned}$$

Thus, the original expression simplifies to:

$$5 \times 5041$$

We have expressed the number as the product of two integers: $5$ and $5041$. Both $5$ and $5041$ are integers greater than $1$. Since the number can be written as a product of two integers other than $1$ and itself, it is by definition a composite number.