Question

# 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.

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Hint: The given number is a composite number if the number has at least one more factor other than 1 and number itself. To prove that the given numbers are composite we have to show that there exists at least one more factor other than $1$ and number itself.

Given numbers are $7 \times 11 \times 13 + 13$ and $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$.
We have to explain that these numbers are composite number
(1) $7 \times 11 \times 13 + 13$
By taking $13$ as common from both the terms $7 \times 11 \times 13 + 13$ can be written as
$= 13 \times \left( {7 \times 11 + 1} \right)$
This clearly shows that there is at least one more factor other than $1$ and number itself that is $13$. So, the given number $7 \times 11 \times 13 + 13$ has more than two factors.
Thus, the given number $7 \times 11 \times 13 + 13$ is a composite number.
(2) $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$
By taking $5$ as common from both the terms $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$ can be written as
$= 5 \times \left( {7 \times 6 \times 4 \times 3 \times 2 \times 1 + 1} \right)$
This clearly shows that there is at least one more factor other than $1$ and the number itself is $5$ so, the given number $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$ has more than two factors.

Thus, the given number $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$ is a composite number.

Note:
Similarly, we can check if the given number is prime or not. If the given number has only two factors that is $1$ and the number itself then the given number is prime number.
Alternatively: These can be solved by finding the number by simply multiplying the numbers and then adding the given number and then find their prime factors and if we got more than one prime factor then the given numbers are composite numbers otherwise prime numbers.