Question

The nature of the numbers 11, 13 is
A. Coprime numbers
B. Prime numbers
C. Both prime and coprime numbers.
D. None of these.

Answer 1

Hint: We have to know the definitions of prime numbers and coprime numbers and the difference between them to solve this problem. A prime number is defined as the number which has only two factors that are 1 and itself. Coprime numbers are defined for two numbers which states that two numbers are coprime if the highest common factor of the two numbers is 1. Using these properties, we can find the relation between 11, 13, and the answer.

Complete step-by-step solution:
In the question, the given numbers are 11, 13. Let us check the prime number property of the two numbers 11, 13.
A number ‘a’ is defined as a prime number if the factors of the number are 1 and ‘a’ itself.
Let us consider the number 11. We can infer that there is no other number that divides 11 with a remainder zero. The factorisation of 11 is as follows
$\begin{align}
  & 11\left| \!{\underline {\,
  11 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
The factorisation of 11 is $11=11\times 1$. So, we can conclude that 11 is the prime number.
Let us consider the number 13. We can infer that there is no other number that divides 13 with a remainder zero. The factorisation of 13 is as follows
$\begin{align}
  & 13\left| \!{\underline {\,
  13 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
The factorisation of 13 is $13=13\times 1$. So, we can conclude that 13 is the prime number.
Let us consider the coprime property of the numbers 11, 13.
Two numbers a, b are said to be coprime if the highest common factor of the two numbers is 1.
Using this property and by finding the highest common factor of the numbers 11, 13, we get-
The factors of 11 are 11 and 1. The factors of 13 are 13 and 1. We can infer that the common factor of the numbers is 1.
So, we can conclude that the two numbers 11, 13 are coprime numbers.
$\therefore $ The numbers 11, 13 are prime numbers and coprime numbers. Option C is the right answer.

Note: From the property that the prime number has factors of 1 and itself and two prime numbers have the highest common factor as 1, we can infer that any two prime numbers are always coprime to each other. Another specialty is that 11 and 13 are twin primes too since they are prime numbers and have a common difference of 2.