Question

Write all prime numbers between 50 and 100.

Answer 1

Hint:
Here, we will be using the various divisibility rules to easily identify the numbers that are not prime, and eliminating them from our answer. A prime number is a number divisible by only 1 and itself. A composite number is a number which is not a prime number, that is it is divisible by 1, itself, and some other numbers as well.

Complete step by step solution:
We know that in the number system, prime numbers are the numbers which have only two factors, 1 and the number itself. Hence, it is obvious that they will be divisible by 1 and the number itself.
Composite numbers are the numbers which are not prime numbers. They are divisible by at least one more factor other than 1 and themselves.
We will use divisibility rules to easily identify the composite numbers between 50 and 100, and then eliminate them from our answer.
First, we check the divisibility by 2.
We know that all even numbers are divisible by 2. Thus, all numbers divisible by 2 (except 2 itself) are composite numbers.
Hence, we can eliminate the numbers 50, 54, 56, 58, …, 94, 96, 98, 100 from our answer.
Now, we check the divisibility by 5.
All numbers that end with the digit 0 or 5 are divisible by 5. Thus, the numbers 50, 55, 60, 65, …, 85, 90, 95, 100 are divisible by 5.
Hence, we can eliminate the numbers 55, 65, 75, 85, 95 from our answer.
The remaining numbers are 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 93, 97, 99.
Next, let us check the divisibility of these numbers by 3.
We sum up the digits of the remaining numbers. The number is divisible by 3 if the sum of digits is divisible by 3.
The sum of the numbers 51, 57, 63, 69, 81, 87, 93, 99 is divisible by 3.
Hence, we remove these numbers from our answer.
The remaining numbers are 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97.
Now, we check the divisibility of the number by 7. To check divisibility by 7, twice the unit’s place digit subtracted from the rest of the number should be divisible by 7.
For example, we observe that \[5 - \left( {3 \times 2} \right) = 5 - 6 = - 1\]. Hence, 53 is not divisible by 7.
Checking the rest of the numbers, we observe that
\[7 - \left( {7 \times 2} \right) = 7 - 14 = - 7\]
\[9 - \left( {1 \times 2} \right) = 9 - 2 = 7\]
Hence, the numbers 77 and 91 are divisible by 7.
We will eliminate these from the remaining numbers.

The remaining numbers are 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
These numbers are the prime numbers between the numbers 50 and 100.

Note:
In this problem, we have used the divisibility rule to find the numbers. Divisibility rule is a way of finding whether a number is completely divisible by another number or not without actually dividing the number. We can easily find the numbers that are prime using the divisibility rule. The given range has 50 possible numbers. Using divisibility tests helps us to eliminate many numbers from the range quickly. After that, we can individually check each of the remaining numbers if needed to see whether they are prime or composite.