I need answer only for question \"d\" answer for a and b : https://www.chegg.com

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I need answer only for question "d"

answer for a and b :

https://www.chegg.com/homework-help/questions-and-answers/part-cryptographic-security-team-military-facility-tasked-decoding-messages-required-messa-q22334135

answer for c :

https://www.chegg.com/homework-help/questions-and-answers/need-answer-question-c-d--someone-already-answered-b-following-link-https-wwwcheggcom-home-q22360933

You are part of the Cryptographic Security Team at a military facility, tasked with decoding messages as required. A message has come in from HQ, as an ordered collection of numbers, which is personalised for yourself and aimed at testing aspects of your decoding skills. 247500, 6592, 82121, 76332, 296445, 647 10, 164949, 206921, 266 199, 3 1572 1, 329 746, 442 869 These numbers (above) have each been encoded using RSA with a modulus of m 496241 (with p and q being primes) and encoding exponent of 218821 You are advised that 13631, 142703 is a valid encoding decoding pair for the same modulus, m (a) Use this information to determine p (m) for this modulus. Using software to directly factorise m is not a valid option for doing this part. (b) Verify your answer by determining the primes p and q Show how these combine to give both m and p(m) (c) Calculate the decoding exponent for 218821, as encoding exponent, using the extended Euclidean algorithm (Again, using software to directly obtain this is not a valid option, though you are welcome to use software to confirm your answer.) (d) For each of the 12 numbers in your message, verify they have no prime factors in common with m (It is OK to use software for this task, provided you have answered the previous part.)

Explanation / Answer

d)

The factors for 496241 are: 733 * 677

Given 12 messages are: 247500, 6592, 82121, 76332, 296445, 64710, 164949, 206921, 266199, 315721, 329746, 442869

1) Factors for 247500 are:

247500
|        
123750   2
|        
61875   2
|        
20625   3
|        
6875   3
|        
1375   5
|        
275   5
|        
55   5
|        
11   5

Prime factors for 247500 are: 247500 = 2 × 2 × 3 × 3 × 5 × 5 × 5 × 5 × 11

There is no prime factor for 247500 common with m (496241)

2)Factor pairs for 6592 are: (1, 6592) (2, 3296) (4, 1648) (8, 824) (16, 412) (32, 206) (64, 103)

Prime factors: 6592 = 2 × 2 × 2 × 2 × 2 × 2 × 103

There is no prime factor for 6592 common with m (496241).

3) Factor pairs for 82121 are: (1, 82121) (13, 6317).
Prime factors: 82121 = 13 × 6317

There is no prime factor for 82121 common with m (496241).

4) Factor pairs for 76332 are: (1, 76332) (2, 38166) (3, 25444) (4, 19083) (6, 12722) (12, 6361)
Prime factors: 76332 = 2 × 2 × 3 × 6361

There is no prime factor for 76332 common with m (496241).

5) Factor pairs for 296445 are: (1, 296445) (3, 98815) (5, 59289) (15, 19763)
Prime factors for 296445 are: 3 × 5 × 19763

There is no prime factor for 296445 common with m (496241).

6) Factor pairs for 64710 are: (1, 64710) (2, 32355) (3, 21570) (5, 12942) (6, 10785) (9, 7190) (10, 6471) (15, 4314) (18, 3595) (30, 2157) (45, 1438) (90, 719)

Prime factors for 64710 are: 2 × 3 × 3 × 5 × 719

There is no prime factor for 64710 common with m (496241).

7)Factor pairs for 164949 are: (1, 164949) (3, 54983)
Prime factors for 164949 are: 3 × 54983

There is no prime factor for 164949 common with m (496241).

8) Factor pairs for 206921 are: (1, 206921) (11, 18811) (13, 15917) (143, 1447)
Prime factors for 206921 are: 11 × 13 × 1447

There is no prime factor for 206921 common with m (496241).

9)Factor pairs for 266199 are: (1, 266199) (3, 88733) (89, 2991) (267, 997)
Prime factors for 266199 are: 3 × 89 × 997

There is no prime factor for 266199 common with m (496241).

10)Factor pairs for 315721 are: (1, 315721) (7, 45103) (23, 13727) (37, 8533) (53, 5957) (161, 1961) (259, 1219)
Prime factors for 315721 are: 7 × 23 × 37 × 53

There is no prime factor for 315721 common with m (496241).

11)Factor pairs for 329746 are: (1, 329746) (2, 164873) (79, 4174) (158, 2087)
Prime factors for 329746 are: 2 × 79 × 2087

There is no prime factor for 329746 common with m (496241).

12) Factor pairs for 442869 are: (1, 442869) (3, 147623) (7, 63267) (21, 21089)
Prime factors for 442869 are: 3 × 7 × 21089

There is no prime factor for 442869 common with m (496241).

Therefore for all given 12 numbers of message there are no prime factor that is common with 'm'

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