Consider the primes p = 13 and q = 19. Let e = 43. (a) Compute pq and (p − 1)(q − 1). Prove that gcd(e, (p − 1)(q − 1)) = 1. Compute the smallest positive integer d such that de 1 (mod (p − 1)(q − 1)) 43d 1 (mod 216) Equivalently,

Q: 4444 is divided by a 1.19) find the remainder when 4444 * 2?=-1 (mod a).

A: 1.19 We have 4444  ≡  7 (mod 9). So, 44444444 ≡ 7 4444 (mod 9) Now,       73 ≡1 (mod 9) Or,…

Q: 1. Show that k1601 = k (mod 17) for any k.

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Q: 1. Prove (2p+1) = 2 (mod p) when p is any prime.

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Q: Solve the congruence 11x = 45 (mod 97) where x is an integer in the interval [0,97).

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Q: Theorem 1. Let r, y, z be positive integers where r is odd and y is even. Then r, y, z is a…

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Q: 2. Let p be a prime number such that p> 5. Prove that p2-1 =0 (mod 24).

A: To prove the below statement If p is a prime number and p > 5. p2−1≡0mod24

Q: (d) 9r = 3 (mod 5) (e) 5x = 1 (mod 6) (f) 3x = 1 (mod 6)

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Q: 3.1.12 Suppose that 7x = 28 (mod 42). By Theorem 3.9, it follows that x = 4 (mod 6). (a) Check this…

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Q: Consider the primes p = 5 and q = 11. Let e = 3. (a) Compute pq and (p − 1)(q − 1). Prove that…

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Q: Solve for d, 19d ≡ 1 mod (37422000) using the "Extended Euclid's Algorithm"

A: The solution is given by using Extended Euclid's Algorithm as follows

Q: Encrypt the message STOP using the RSA cryptosystem with key (2537, 13). Note that 2537 = 43 ⋅ 59, p…

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Q: 11.5. For each part, find an x that solves the given simultaneous congruences. (a) x = 3 (mod 7) and…

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Q: 9. Determine which of the following congruences have solutions, and find any solutions (between 0…

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Q: 8. Show that 18!= -1 (mod 437). 9. Using Euler's Theorem, find the last two digits of 44443333 10.…

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