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**Hint**: In this type of question, students should remember the definition of the term ‘twin prime’. Twin prime are those prime numbers which have a gap of 2 in between them. A prime number is a number which has only 2 factors i.e. 1 and the numbers itself. Let us apply the concepts of prime numbers and twin prime numbers to solve the given question.

**:**

__Complete step-by-step answer__In the question, a list of prime numbers from 100 to 200 is given.

They are as follows: 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199.

Let us check the difference between these prime numbers.

Consider the numbers 101 and 103. The difference between 103 and 101 is 2. So, $ \left( {101,103} \right) $ is a pair of twin prime numbers.

Consider the numbers 103 and 107. The difference between 103 and 107 is 4. So, $ \left( {107,103} \right) $ is not a pair of twin prime numbers.

Consider the numbers 107 and 109. The difference between 109 and 107 is 2. So, $ \left( {107,109} \right) $ is a pair of twin prime numbers.

Consider the numbers 109 and 113. The difference between 109 and 113 is 4. So, $ \left( {109,113} \right) $ isn’t a pair of twin prime numbers.

Consider the numbers 113 and 127. The difference between 113 and 127 is 14. So, $ \left( {113,127} \right) $ isn’t a pair of twin prime numbers.

Consider the numbers 127 and 131. The difference between 127 and 131 is 4. So, $ \left( {127,131} \right) $ isn’t a pair of twin prime numbers.

Consider the numbers 137 and 131. The difference between 137 and 131 is 6. So, $ \left( {137,131} \right) $ isn’t a pair of twin prime numbers.

Consider the numbers 137 and 139. The difference between 137 and 139 is 2. So, $ \left( {137,139} \right) $ is a pair of twin prime numbers.

Consider the numbers 139 and 149. The difference between 139 and 149 is 10. So, $ \left( {139,149} \right) $ isn’t a pair of twin prime numbers.

Consider the numbers 149 and 151. The difference between 149 and 151 is 2. So, $ \left( {149,151} \right) $ is a pair of twin prime numbers.

Consider the numbers 157 and 151. The difference between 157 and 151 is 6. So, $ \left( {157,151} \right) $ isn’t a pair of twin prime numbers.

Consider the numbers 157 and 163. The difference between 157 and 163 is 6. So, $ \left( {157,163} \right) $ isn’t a pair of twin prime numbers.

Consider the numbers 167 and 163. The difference between 167 and 163 is 4. So, $ \left( {167,163} \right) $ isn’t a pair of twin prime numbers.

Consider the numbers 167 and 173. The difference between 167 and 173 is 6. So, $ \left( {167,173} \right) $ isn’t a pair of twin prime numbers.

Consider the numbers 179 and 173. The difference between 179 and 173 is 6. So, $ \left( {179,173} \right) $ isn’t a pair of twin prime numbers.

Consider the numbers 179 and 181. The difference between 179 and 181 is 2. So, $ \left( {179,181} \right) $ is a pair of twin prime numbers.

Consider the numbers 191 and 181. The difference between 191 and 181 is 10. So, $ \left( {191,181} \right) $ isn’t a pair of twin prime numbers.

Consider the numbers 191 and 193. The difference between 191 and 193 is 2. So, $ \left( {191,193} \right) $ is a pair of twin prime numbers.

Consider the numbers 197 and 193. The difference between 197 and 193 is 4. So, $ \left( {197,193} \right) $ isn’t a pair of twin prime numbers.

Consider the numbers 197 and 199. The difference between 197 and 199 is 2. So, $ \left( {197,199} \right) $ is a pair of twin prime numbers.

Therefore, the pairs of twin-prime numbers are $ \left( {101,103} \right) $ , $ \left( {107,109} \right) $ , $ \left( {137,139} \right)$ , $ \left( {149,151} \right) $ , $ \left( {179,181} \right) $ , $ \left( {191,193} \right) $ , $ \left( {197,199} \right) $ .

**So, the correct answer is “ $ \left( {101,103} \right) $ , $ \left( {107,109} \right) $ , $ \left( {137,139} \right)$ , $ \left( {149,151} \right) $ , $ \left( {179,181} \right) $ , $ \left( {191,193} \right) $ , $ \left( {197,199} \right) $ .”.**

**Note**: Students often make the mistake of considering a single number as a pair of twin prime numbers. A pair of twin prime numbers consists of two numbers. Students must take utmost care while calculating the difference between the prime numbers and categorise the pair of prime numbers as twin prime numbers only when the difference between the numbers is 2

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