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?

Answer 1

Hint: Try to convert the numbers into $a \times b$ format
A composite number is any number that can be expressed in $a \times b$ form. So our approach should be to convert the given sequence into the required sequence of $a \times b$. Multiplying all the numbers first, we will then take the common terms out to convert these numbers into the $a \times b$f orm and prove that both the numbers are composite.

Complete step-by-step answer:
In order to solve problems like this
The very first step we need to do is to know what is a composite number.
So, a composite number is a number which has more than one factor.
Mathematically speaking, a composite number is a number which can be written in the form of $a \times b$ format where neither a nor b are either 0 or the number itself.
So, taking the first question we see that
$7 \times 11 \times 13 + 13$ is divided into two parts separated by addition sign
Now, if we simplify it, we will get
$7 \times 11 \times 13 + 13 = 77 \times 13 + 13$
Now, if we take 13 common from both the terms we will get
$77 \times 13 + 13 = 13 \times \left( {77 + 1} \right)$
If we consider$a = 13$,$b = \left( {77 + 1} \right)$
$13 \times \left( {77 + 1} \right) = a \times b$
Hence, the given number can be written in the form of $a \times b$
Therefore, we can conclude that $7 \times 11 \times 13 + 13$ is composite.
Now, taking the second part
$7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$ is divided into two parts separated by addition sign
Now, if we simplify it, we will get
$7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5 = 5040 + 5$
Now, if we take 5 common from both the terms we will get
$5040 + 5 = 5\left( {1008 + 1} \right)$
If we consider$a = 5$,$b = \left( {1008 + 1} \right)$
$5 \times \left( {1008 + 1} \right) = a \times b$
Hence, the given number can be written in the form of $a \times b$
Therefore, we can conclude that $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$ is composite.

Note: Do not add the numbers first and then multiply the rest of the numbers. In this way, you will not only break the BODMAS rule, but also the answer will not be correct. Hence, multiply all the numbers first and then take the common term outside in order to solve these types of questions.