Question

How many natural numbers are there between 102 and 112? What fraction of them are prime numbers?

Answer 1

Hint:
Here, we need to find the natural numbers between 102 and 112, and determine what fraction of these are prime numbers. First, we will write down the natural numbers between 102 and 112, and count them. Then, we will find which of the numbers that are prime by using divisibility tests. Finally, we will express the number of prime numbers between 102 and 112 as a fraction.

Complete step by step solution:
A natural number is any integer that comes after 0.
For example, 1, 2, 3, 4, 143, etc. are all natural numbers.
We will write down the natural numbers between 102 and 112. The natural numbers between 102 and 112 are 103, 104, 105, 106, 107, 108, 109, 110, 111.
Therefore, there are 9 natural numbers between 102 and 112.
Now, we will find the prime numbers between 102 and 112. A prime number is a number which is divisible only by 2 natural numbers, that is by 1 and the number itself.
For example: 5 is a prime number because it is divisible by only 2 natural numbers, 1 and 5.
We will check the divisibility of the numbers 103, 104, 105, 106, 107, 108, 109, 110, 111 by 2, 3, 5, etc. to find which of them are not prime numbers.
First, let us check the divisibility by 2.
We know that every even number is divisible by 2. This means that any number that has any of the digits 0, 2, 4, 6, 8 in the units place is divisible by 2.
Therefore, 104, 106, 108, 110 are divisible by 2. Thus, these are not prime numbers.
The remaining numbers are 103, 105, 107, 109, 111.
Now, we will check the divisibility by 3.
The number is said to be divisible by 3 only if the sum of the digits of a number is divisible by 3.
We can see that \[1 + 0 + 5 = 6\] and \[1 + 1 + 1 = 3\].
Since 6 and 3 are divisible by 3, 105 and 111 are divisible by 3.
Thus, these are not prime numbers.
The remaining numbers are 103, 107, 109.
103, 107, and 109 are prime numbers since they are not divisible by any number other than 1 or itself.
Therefore, there are 3 prime numbers in the 9 natural numbers between 102 and 112, that is 103, 104, 105, 106, 107, 108, 109, 110, 111.
We need to express this as a fraction.
A fraction is a number which represents a part of a group. It is written as \[\dfrac{a}{b}\], where \[a\] is called the numerator and \[b\] is called the denominator. The group is divided into \[b\] equal parts. The fraction \[\dfrac{a}{b}\] shows that \[a\] out of \[b\] equal parts of the group.
We can observe that in the group of 9 numbers, 3 numbers are prime.
This can be expressed as a fraction with numerator 3 and denominator 9.
Therefore, the number of prime natural numbers between 102 and 112 can be written as the fraction \[\dfrac{3}{9}\].
Simplifying this fraction, we get
\[ \Rightarrow \dfrac{3}{9} = \dfrac{1}{3}\]

Therefore, the number of prime natural numbers between 102 and 112 can be expressed as the fraction \[\dfrac{3}{9}\], or \[\dfrac{1}{3}\].

Note:
Here, we can make a mistake by including 102 and 112 in the list of natural numbers. This will be incorrect because the question states to find the numbers between 102 and 112. Unless specified, you should always assume 102 and 112 to be excluded.