Question

Write all the primes less than $25$, to which number set the square roots of these prime numbers belongs to?

Answer 1

Hint: To solve the question, we have to use the concept of the prime numbers condition which states a prime number has only two factors, $1$ and the number itself. List down all the natural numbers below $25$ and apply the divisibility rule of \[2,3,5,7\] to eliminate the non-prime numbers. Thus, can we obtain the list of prime numbers below $25$.

Complete step by step answer:
In the first part of the question, we have to find all the prime numbers less than $25$.
So,
The numbers below $25$ are \[1,{\text{ }}2,{\text{ }}3,{\text{ }}4,{\text{ }}5,{\text{ }}6,{\text{ }}7,{\text{ }}8,{\text{ }}9,{\text{ }}10,{\text{ }}11,{\text{ }}12,{\text{ }}13,{\text{ }}14,{\text{ }}15,{\text{ }}16,{\text{ }}17,{\text{ }}18,{\text{ }}19,{\text{ }}20,{\text{ }}21,{\text{ }}22,{\text{ }}23,{\text{ }}24.\]
A prime number is a number which has only two factors $1$ and itself. Thus, we can eliminate the number $1$ since it has only factor $1$.
The remaining numbers below $25$ are \[2,{\text{ }}3,{\text{ }}4,{\text{ }}5,{\text{ }}6,{\text{ }}7,{\text{ }}8,{\text{ }}9,{\text{ }}10,{\text{ }}11,{\text{ }}12,{\text{ }}13,{\text{ }}14,{\text{ }}15,{\text{ }}16,{\text{ }}17,{\text{ }}18,{\text{ }}19,{\text{ }}20,{\text{ }}21,{\text{ }}22,{\text{ }}23,{\text{ }}24.\]
Among the given numbers even numbers greater than $2$ below $25$ are
\[4,{\text{ }}6,{\text{ }}8,{\text{ }}9,{\text{ }}10,{\text{ }}12,{\text{ }}14,{\text{ }}16,{\text{ }}18,{\text{ }}20,{\text{ }}21,{\text{ }}22,{\text{ }}24.\]
All the even numbers greater than $2$ have factors $2$ other than the number and $1$. Thus, they cannot be termed as prime numbers.
The remaining numbers are \[2,{\text{ }}3,{\text{ }}5,{\text{ }}7,{\text{ }}9,{\text{ }}11,{\text{ }}13,{\text{ }}15,{\text{ }}17,{\text{ }}19,{\text{ }}21,{\text{ }}23\].
Among the remaining numbers \[9,{\text{ }}15,{\text{ }}21\] are divisible by $3$.
Thus, they cannot be termed as prime numbers.
The remaining numbers are \[2,{\text{ }}3,{\text{ }}5,{\text{ }}7,{\text{ }}11,{\text{ }}13,{\text{ }}17,{\text{ }}19,{\text{ }}23\].
$\therefore$ The prime numbers below $25$ are \[2,{\text{ }}3,{\text{ }}5,{\text{ }}7,{\text{ }}11,{\text{ }}13,{\text{ }}17,{\text{ }}19,{\text{ }}23\].
Now, in the second part of the question we have to find the set of the square roots of these prime numbers.
As the prime numbers are not the multiple of any numbers. Therefore, the square root of all the prime numbers are Irrational numbers.

Note:
The possibility of mistake can be in not applying all the divisibility rules of single digit prime numbers- \[2,3,5,7\] to eliminate all the factors of these numbers to narrow down them to a list of prime numbers. The other possibility of mistake can be not rechecking the remaining list of numbers for further confirmation.