11. Categorise the given pairs as co-prime or twin
primes.
(a) (41, 43) (b) (45, 47) (c) (49, 51)
(d) (51, 53) (e) (55, 57) (f) (63, 65)
(g) (67, 69) (h) (71, 73)
Answers
Topic :-
Number System
Given :-
(a) (41, 43) (b) (45, 47) (c) (49, 51) (d) (51, 53) (e) (55, 57) (f) (63, 65) (g) (67, 69) (h) (71, 73)
Question :-
Categorise the given pairs as co-prime or twin primes.
Solution :-
Co-prime Numbers
Two natural numbers (not necessarily prime) are co-prime, if their H.C.F. is 1.
Twin Prime Numbers
If the difference between two prime numbers is two, then the numbers are twin prime numbers.
Checking given pairs,
(a) (41, 43)
43 - 41 = 2
41 and 43 are prime numbers.
41 and 43 have H.C.F. = 1
Hence, this pair is Co-prime as well as Twin Prime too.
(b) (45, 47)
47 - 45 = 2 but
45 is not a prime number.
45 and 47 have H.C.F. = 1
Hence, this pair is Co-Prime.
(c) (49, 51)
51 - 49 = 2 but
49 and 51 are not prime numbers.
49 and 51 have H.C.F. = 1
Hence, this pair is Co-Prime.
(d) (51, 53)
53 - 51 = 2 but
51 is not a prime number.
51 and 53 have H.C.F. = 1
Hence, this pair is Co-prime.
(e) (55, 57)
57 - 55 = 2 but
55 and 57 are not prime numbers.
55 and 57 have H.C.F. = 1
Hence, this pair is Co-Prime.
(f) (63, 65)
65 - 63 = 2 but
63 and 65 are not prime numbers.
63 and 65 have H.C.F. = 1
Hence, this pair is Co-Prime.
(g) (67, 69)
69 - 67 = 2 but
69 is not a prime number.
67 and 69 have H.C.F. = 1
Hence, this pair is Co-Prime.
(h) (71, 73)
73 - 71 = 2
71 and 73 are prime numbers.
71 and 73 have H.C.F. = 1
Hence, this pair is Co-prime as well as Twin Prime too.
Answer :-
Co-prime as well as Twin-Prime :- (a) and (h)
Co-Prime :- (b), (c), (d), (e), (f) and (g)