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Question

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(a) (19, 21)

(b) (29, 31)

(c) (39, 41)

(d) (49, 51)

Answer

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Hint: First of all check if both the numbers are prime or not. If yes, then check if the difference between the numbers is 2 or not. If both conditions satisfy, then only two numbers are twin prime.

Complete step-by-step answer:

Here, we have to find which of the following pairs twin primes are: (19, 21), (29, 31), (39, 41), (49, 51).

Before proceeding with this question, we must know what twin prime numbers are. Two numbers are twin prime to each other when they both are prime and have a difference of 2 between them. For example, (5, 7) and (11, 13), etc. are twin prime numbers. Also, a prime number is a number that is divisible by 1 and itself only. For example, 2, 7, 17, etc. are prime numbers.

Now, we will check which of the pairs in options are twin prime.

(a) (19, 21)

In the above pair, 19 is a prime number, and the difference between 19 and 21 is 2, but 21 is not a prime number as it has factors like 3, 7, etc. So, we get that the pair (19, 21) is not a twin prime.

(b) (29, 31)

In the above pair, both 29 and 31 are prime numbers and the difference between 29 and 31 is 2. So, we get that the pair (29, 31) is twin prime.

(c) (39, 41)

In the above pair, 41 is a prime number but 39 is not a prime number as it has factors like 3, 13, etc. So we get that the pair (39, 41) is not a twin prime.

(d) (49, 51)

In the above pair, neither 49 nor 51 is a prime number because 49 has factors like 7, etc. and 51 has factors like 17, 3, etc. So, we get that the pair (49, 51) is not a twin prime.

So, we get only one pair that is twin prime and that pair is (29, 31).

Hence, option (b) is the right answer.

Note: Students must note that 2 is the smallest prime number and is the only even prime number. All other prime numbers are odd. Students often make this mistake of considering many odd numbers as prime. For example, many students consider 237, 147, 51, 91, etc. as prime by just looking at them, but they are not prime numbers. So, care must be taken while selecting numbers as prime or not.

Complete step-by-step answer:

Here, we have to find which of the following pairs twin primes are: (19, 21), (29, 31), (39, 41), (49, 51).

Before proceeding with this question, we must know what twin prime numbers are. Two numbers are twin prime to each other when they both are prime and have a difference of 2 between them. For example, (5, 7) and (11, 13), etc. are twin prime numbers. Also, a prime number is a number that is divisible by 1 and itself only. For example, 2, 7, 17, etc. are prime numbers.

Now, we will check which of the pairs in options are twin prime.

(a) (19, 21)

In the above pair, 19 is a prime number, and the difference between 19 and 21 is 2, but 21 is not a prime number as it has factors like 3, 7, etc. So, we get that the pair (19, 21) is not a twin prime.

(b) (29, 31)

In the above pair, both 29 and 31 are prime numbers and the difference between 29 and 31 is 2. So, we get that the pair (29, 31) is twin prime.

(c) (39, 41)

In the above pair, 41 is a prime number but 39 is not a prime number as it has factors like 3, 13, etc. So we get that the pair (39, 41) is not a twin prime.

(d) (49, 51)

In the above pair, neither 49 nor 51 is a prime number because 49 has factors like 7, etc. and 51 has factors like 17, 3, etc. So, we get that the pair (49, 51) is not a twin prime.

So, we get only one pair that is twin prime and that pair is (29, 31).

Hence, option (b) is the right answer.

Note: Students must note that 2 is the smallest prime number and is the only even prime number. All other prime numbers are odd. Students often make this mistake of considering many odd numbers as prime. For example, many students consider 237, 147, 51, 91, etc. as prime by just looking at them, but they are not prime numbers. So, care must be taken while selecting numbers as prime or not.