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Question

Answers

A. $16$ and $26$, $19$ and $29$

B. $29$ and $31$, $41$ and $43$

C. $11$ and $21$, $61$ and $71$

D. $15$ and $17$, $37$ and $39$

Answer
Verified

Hint: In order to deal with this question first we have to define the term twin prime numbers further we will discuss its properties and according to it we will determine the required prime pair.

Complete step-by-step answer:

Two prime numbers are considered twin primes, if only one composite number is present between them. Or we can also call twin primes, two prime numbers whose difference is two. For instance, $(3,5)$ are twin primes because the difference between the two numbers is $5 - 3 = 2$. The alternate names given to twin primes are twin or prime pair twins.

Prime pair key properties:

• $(2,3)$ are not known to be twin primes, because there is no composite number between them, and the difference between the two primes is not $2$.

• $5$ is a prime number only available in two separate pairs

Any other prime pair$(3,5)$ is in the form of $(6n - 1,6n + 1),$where n is any natural number.

• The sum of each prime pair is divisible by $12$, apart from $(3,5)$.

By considering option B

$29$ and $31$, $41$ and $43$

In both the pairs, difference between two numbers is \[31 - 29{\text{ }} = {\text{ }}43 - 41 = {\text{ }}2\]

And the sum of each prime pair \[29 + 31{\text{ }} = {\text{ }}60\] and \[41 + 43{\text{ }} = {\text{ }}84\] is divisible by $12$

Therefore it follows the above property of prime pair

Hence the correct answer is option B.

Note- A twin prime is a prime number that is either $2$ less or $2$ more than another prime number — either member of the twin prime pair, for example. In other words, a twin prime is a prime with a prime difference of two.

Complete step-by-step answer:

Two prime numbers are considered twin primes, if only one composite number is present between them. Or we can also call twin primes, two prime numbers whose difference is two. For instance, $(3,5)$ are twin primes because the difference between the two numbers is $5 - 3 = 2$. The alternate names given to twin primes are twin or prime pair twins.

Prime pair key properties:

• $(2,3)$ are not known to be twin primes, because there is no composite number between them, and the difference between the two primes is not $2$.

• $5$ is a prime number only available in two separate pairs

Any other prime pair$(3,5)$ is in the form of $(6n - 1,6n + 1),$where n is any natural number.

• The sum of each prime pair is divisible by $12$, apart from $(3,5)$.

By considering option B

$29$ and $31$, $41$ and $43$

In both the pairs, difference between two numbers is \[31 - 29{\text{ }} = {\text{ }}43 - 41 = {\text{ }}2\]

And the sum of each prime pair \[29 + 31{\text{ }} = {\text{ }}60\] and \[41 + 43{\text{ }} = {\text{ }}84\] is divisible by $12$

Therefore it follows the above property of prime pair

Hence the correct answer is option B.

Note- A twin prime is a prime number that is either $2$ less or $2$ more than another prime number — either member of the twin prime pair, for example. In other words, a twin prime is a prime with a prime difference of two.

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