Question

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

Answer 1

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
$$\begin{gathered} \Rightarrow 15\left(7\times 11\times 13\times 1+1\right) \\ \Rightarrow 15\left(1001+1\right) \\ \Rightarrow 15\left(1002\right) \end{gathered}$$
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.