Question

If \[7 \times 11 \times 13 + 13\] is a composite number. Justify the statement.

Answer 1

Hint:
Composite number: The composite numbers can be defined as the whole numbers that have more than two factors. Every whole number or positive integer has at least two factors (One factor will be 1 and another factor will be itself).

Complete step by step solution:
Given number: \[7 \times 11 \times 13 + 13\]
Simplify the given number
\[
  7 \times 11 \times 13 + 13 \\
   \Rightarrow 13\left( {7 \times 11 + 1} \right) \\
   \Rightarrow 13\left( {77 + 1} \right) \\
   \Rightarrow 13\left( {78} \right) \\
\]
Let the product of the number be (P)
\[ \Rightarrow \left( P \right) = 13\left( {78} \right)\]
A composite number is a positive integer that has more than two factors. One factor will be 1 and the other factor will be itself. If any number has any factor (excluding the above mentioned) two factors, then the number will be a composite number else a prime number.
But here we can clearly see that the number has two other factors \[13\] and \[78\] other than 1 and itself \[\left( P \right)\]

Hence, we can conclude that the given number \[7 \times 11 \times 13 + 13\] is a composite number.

Note:
If any number \[\left( N \right)\] can be written \[\left( N \right) = m \times n\]; where \[m,n \ne 1\], will have \[m\] and \[n\] as factors of that number other than 1 and itself. This type of numbers will always form composite numbers