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: In this question, we use the concept of composite number. Composite numbers can be defined as the whole numbers that have more than two factors. Basically, Composite numbers are the numbers which have factors other than 1 and the number itself.

Complete step-by-step answer:

Now, we have to explain $7 \times 11 \times 13 + 13$ and $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$ are composite numbers.
So, we have to prove a given number has more than two factors (have factors other than 1 and the number itself).
Now,
$
   \Rightarrow 7 \times 11 \times 13 + 13 \\
   \Rightarrow 13\left( {7 \times 11 + 1} \right) \\
   \Rightarrow 13\left( {77 + 1} \right) \\
   \Rightarrow 13 \times 78 \\
   \Rightarrow 13 \times 13 \times 6 \\
   \Rightarrow 13 \times 13 \times 3 \times 2 \\
$
We can see that given numbers have more than two factors (other than 1 and number itself). So, $7 \times 11 \times 13 + 13$ number is a composite number.
 Now,
$
   \Rightarrow 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5 \\
   \Rightarrow 5\left( {7 \times 6 \times 4 \times 3 \times 2 \times 1 + 1} \right) \\
   \Rightarrow 5\left( {1008 + 1} \right) \\
   \Rightarrow 5 \times 1009 \\
$
 We can see given numbers have more than two factors (other than 1 and number itself). So, $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$ number is a composite number.
So, it's proved $7 \times 11 \times 13 + 13$ and $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$ are composite numbers.

Note: Whenever we face such types of problems we use some important points. First we factories the given number into more than two factors (other than 1 and number itself). So, any number that has more than two factors (other than 1 and number itself) are called composite numbers.