Q6:

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

Given: Two numerical expressions:
1. $A = 7 \times 11 \times 13 + 13$
2. $B = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$

To Prove: Both $A$ and $B$ are composite numbers.

Theoretical Background:
According to the Fundamental Theorem of Arithmetic, every composite number can be expressed (factorized) as a product of primes. A composite number is defined as a positive integer greater than $1$ that has at least one divisor other than $1$ and itself. In other words, if a number can be expressed as a product of two or more factors where each factor is greater than $1$, it is a composite number.

Step 1: Analyzing the first expression ($A$)
Let $A = 7 \times 11 \times 13 + 13$.
We observe that $13$ is a common factor in both terms of the expression.
Factoring out $13$:
$A = 13 \times (7 \times 11 \times 1 + 1)$ [Using the Distributive Property: $ab + ac = a(b+c)$]
$A = 13 \times (77 + 1)$
$A = 13 \times 78$
Since $78$ can be further factorized as $2 \times 3 \times 13$, we have:
$A = 13 \times 2 \times 3 \times 13 = 2 \times 3 \times 13^2$
Since $A$ is expressed as a product of prime factors ($2, 3, 13$), it satisfies the definition of a composite number.

Step 2: Analyzing the second expression ($B$)
Let $B = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$.
We observe that $5$ is a common factor in both terms of the expression.
Factoring out $5$:
$B = 5 \times (7 \times 6 \times 4 \times 3 \times 2 \times 1 + 1)$ [Using the Distributive Property]
Calculating the product inside the parentheses:
$7 \times 6 = 42$
$42 \times 4 = 168$
$168 \times 3 = 504$
$504 \times 2 = 1008$
$B = 5 \times (1008 + 1)$
$B = 5 \times 1009$
Since $1009$ is a prime number, $B$ is expressed as the product of two factors ($5$ and $1009$), both of which are greater than $1$. Therefore, $B$ is a composite number.

Conclusion:
Since both expressions $A$ and $B$ can be written as a product of factors other than $1$ and the numbers themselves, they are confirmed to be composite numbers.

Final Answer: Both $7 \times 11 \times 13 + 13$ and $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$ are composite numbers because they possess factors other than $1$ and themselves.