\[7 \times 11 \times 13 \times 15 + 15\] is a prime number. Is it true? Justify.

Hint:
Prime number: A prime number is a natural number greater than 1 that has no positive integer divisors other than 1 and itself. For example, 7 is a prime number because it has no positive divisors other than 1 and 7 itself.

Complete step by step solution:
Given number: \[7 \times 11 \times 13 \times 15 + 15\]
Simplify the given number
\[
   \Rightarrow 15\left( {7 \times 11 \times 13 \times 1 + 1} \right) \\
   \Rightarrow 15\left( {1001 + 1} \right) \\
   \Rightarrow 15\left( {1002} \right) \\
\]
Let the product of the number be (P)
\[ \Rightarrow \left( P \right) = 15\left( {1002} \right)\]
A prime number is a positive integer that has two factors. One factor will be 1 and another factor will be the number itself. If any number has exactly two factors, then the number will be a prime number else a composite number.
But here we can clearly see that the number has two other factors \[15\] and \[1002\] 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 and not a prime 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.