Question 1 :
What are twin prime numbers? Give examples.
Question 2 :
How many pairs of twin prime numbers are there between 1 and 100? Write them all.
Question 3 :
How many pairs of twin prime numbers are there between 100 and 200? Write them all.
Question 4 :
How many pairs of twin prime numbers are there between 200 and 300? Write them all.
Question 5 :
How many pairs of twin prime numbers are there between 300 and 400? Write them all.
Question 6 :
How many pairs of twin prime numbers are there between 400 and 500? Write them all.
Question 7 :
How many pairs of twin prime numbers are there between 500 and 1000? Write them all.
Question 8 :
How many pairs of twin prime numbers are there between 1 and 1000?
Question 9 :
Can 3 and 5 be twin prime numbers? Explain.
Question 10 :
Can 2 and 3 be twin prime numbers? Explain.
Question 11 :
Can 31 and 37 be twin prime numbers? Explain.
Question 12 :
The difference between 1 and 3 is 2. Can 1 and 3 be twin prime numbers? Justify your answer.
Question 13 :
Apart from (3, 5), the sum of two primes numbers in each pair of twin primes is divisible by what number? Give examples.
Question 14 :
For a natural number a, can (6a - 1) be a prime number? If so, give some examples.
Question 15 :
For a natural number b, can (6b + 1) be a prime number? If so, give some examples.
Question 16 :
For a natural number k, are all twin primes in the form of (6k - 1, 6k + 1)? If so, give some examples.
Question 17 :
For any natural number k, is (6k - 1, 6k + 1) a twin prime?
1. Answer :
Two prime numbers with a difference of 2 are called twin prime numbers.
Examples :
(3, 5) and (17, 19)
2. Answer :
There are eight pairs of twin prime numbers between 1 and 100.
They are
(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73)
3. Answer :
There are seven pairs of twin prime numbers between 100 to 200.
They are
(101, 103), (107, 109), (137, 139), (149, 151), (179, 181), (191, 193), (197, 199)
4. Answer :
There are four pairs of twin prime numbers between 200 to 300.
They are
(227, 229), (239, 241), (269, 271), (281, 283)
5. Answer :
There are two pairs of twin prime numbers between 300 to 400.
They are
(311, 313), (347, 349)
6. Answer :
There are three pairs of twin prime numbers between 400 and 500.
They are
(419, 421), (431, 433), (461, 463)
7. Answer :
There are eleven pairs of twin prime numbers between 500 and 1000.
They are
(521, 523), (569, 571), (599, 601), (617, 619), (641, 643), (659, 661), (809, 811), (821, 823), (827, 829), (857, 859), (881, 883)
8. Answer :
There are 35 pairs of twin prime numbers between 1 and 1000.
9. Answer :
We know that the two prime numbers with difference 2 are called twin prime numbers.
Both 3 and 5 are prime numbers.
3 - 5 = 2
The difference between 3 and 5 is 2.
Therefore, 17 and 19 are twin prime numbers.
10. Answer :
Both 2 and 3 are prime numbers.
3 - 2 = 1 ≠ 2
Even though 2 and 3 are prime numbers, the difference between them is not 2.
Therefore, 2 and 3 are NOT twin prime numbers.
11. Answer :
Both 31 and 37 are prime numbers.
37 - 31 = 6 ≠ 2
Even though 31 and 31 are prime numbers, the difference between them is not 2,
Thereforte, 31 and 37 are NOT twin prime numbers.
12. Answer :
Even though the difference between 1 and 3 is 2, they can NOT be twin prime numbers. Because, 1 is neither prime nor composite.
13. Answer :
Apart from (3, 5), the sum of two primes numbers in each pair of twin primes is divisible by 12.
Examples :
(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73)
(11, 13) ----> 11 + 13 = 24 (divisible by 12)
(29, 31) ----> 29 + 31 = 60 (divisible by 12)
(71, 73) ----> 71 + 73 = 144 (divisible by 12)
14. Answer :
For a natural number a, (6a - 1) can be a prime number.
Examples :
a = 2 :
6(2) - 1 = 12 - 1 = 11 (prime number)
a = 5 :
6(5) - 1 = 30 - 1 = 29 (prime number)
a = 8 :
6(8) - 1 = 48 - 1 = 47 (prime number)
15. Answer :
For a natural number b, (6b + 1) can be a prime number.
Examples :
b = 3 :
6(3) + 1 = 18 + 1 = 19 (prime number)
b = 6 :
6(6) + 1 = 36 + 1 = 37 (prime number)
b = 10 :
6(10) + 1 = 60 + 1 = 61 (prime number)
16. Answer :
For a natural number k, all twin primes are in the form of
(6k - 1, 6k + 1)
Examples :
Consider the twin prime (5, 7).
5 is 1 less than 6
7 is 1 more than 6
So, (5, 7) is in the form of (6n - 1, 6n + 1).
Consider the twin prime (17, 19).
17 is 1 less than 18, which is a multiple of 6
19 is 1 more than 18, which is a multiple of 6
So, (17, 19) is in the form of (6n - 1, 6n + 1).
17. Answer :
For any natural number k, (6k - 1, 6k + 1) will always not result a twin prime.
Example :
For k = 6,
(6k - 1, 6k + 1) = (35, 37)
Here, (35, 37) is not a twin prime. Because 35 is not a prime number, it is a composite number.
For any natural number k, (6k - 1, 6k + 1) is NOT a twin prime.
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