Question

(i) Express the following as sums of twin primes:
(a) 36
(b) 84
(c) 120
(ii) List all the prime numbers between 50 and 100
(iii) List all the prime numbers between 100 and 150

Answer 1

Hint: In this question, we need to determine the sum of the twin primes of the given numbers. For this, we will evaluate the prime factors of the given numbers and then, add them. In the second part and the third part of the question we will determine all the prime numbers between the given ranges.

Complete step-by-step answer:
Sum of twin primes
a.36
Let x be the number which is the half of the given number, so we can write
\[\Rightarrow x = \dfrac{{36}}{2} = 18\]
Now 18 is the half of the given number, now to find the twin prime number which show have a difference of 2 or more, we can write the sum of twin primes as
\[\left( {19 - 1} \right)\left( {17 + 1} \right) = 36\]{For the difference of 2}
Hence by further solving this equation we get
\[
\Rightarrow 19 + 17 + 1 - 1 = 36 \\
\Rightarrow 19 + 17 = 36 \\
 \]
 Hence the twin prime of 36 is 19 and 17
So, the correct answer is “19 and 17”.

b.84
Let x be the number which is the half of the given number
\[\Rightarrow x = \dfrac{{84}}{2} = 42\]
Following the same method as above, we can write the sum of twin primes as
\[\left( {43 - 1} \right) + \left( {41 + 1} \right) = 84\]{For the difference of 2}
Hence by further solving this equation we get
\[
\Rightarrow 43 + 41 + 1 - 1 = 84 \\
\Rightarrow 43 + 41 = 84 \\
 \]
 Hence the twin prime of 84 is 43 and 41
So, the correct answer is “43 and 41”.

c.120
Let x be the number which is the half of the given number
\[\Rightarrow x = \dfrac{{120}}{2} = 60\]
Following the same method as above, we can write the sum of twin primes as
\[\left( {61 - 1} \right) + \left( {59 + 1} \right) = 120\]{For the difference of 2}
Hence by further solving this equation we get
\[
\Rightarrow 61 + 59 + 1 - 1 = 120 \\
\Rightarrow 61 + 59 = 120 \\
 \]
Hence the twin prime of 120 is 61 and 59.
So, the correct answer is “61 and 59”.

(ii) List all the prime numbers between 50 and 100
All the prime numbers which are between the 50 and 100 and are divisible by 1 or itself are 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
So, the correct answer is “53, 59, 61, 67, 71, 73, 79, 83, 89, 97”.

(iii) List all the prime numbers between 100 and 150
All the prime numbers which are between 100 and 150 and are divisible by 1 or itself are 101, 103, 107, 109, 113, 127, 131, 137, 139, 149.
So, the correct answer is “101, 103, 107, 109, 113, 127, 131, 137, 139, 149”.

Note: Twin prime numbers refers to the prime numbers that are either 2 less or 2 more than another prime number. Prime numbers are generally referred to as the number which can only be divided by the integer 1 and by itself. These numbers are greater than 1 but not the product of small numbers. When we factorize a prime number, then they will only have two factors 1 and the number itself.