We have seen that for every odd prime p the quadratic congruence ²-1=0 (mod p) has exactly two solutions x = ±1 (mod p). Show that if p is of the form p = 3k + 2, fo some k E N, then the cubic congruence 2³-1=0 (mod p) has the single solution z = 1 (mod p).

Q: (1) Determine the number Sm of solutions to the equation xyz = 1 (mod m) for each positive integer…

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Q: The remainder when a is divided by 7 is 2 and the remainder when b is divided by 7 is 5. (So a mod 7…

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Q: a) Let p be an odd prime. Show that a = -1 mod p has a solution if and only if p = 1 mod 8.

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A:

Q: (9) Find all prime numbers p for which both equations X²-2 0 mod p and have solutions. X²-3 0 mod p

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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: (d) How many of the 100 numbers 213! + 2, 213! +3,..., 213! + 101 are composite?

A:

Q: 3. Solve the following congruence equations. (a) 20 984 (mod 1984). 11r + 5y = 7 (mod 20) 6x + 3y =…

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Q: a) Find all solutions to the congruence 10r = 5 mod 15

A: Given,

Q: (2) (a) Find all of the solutions modulo 15 to 6x = 15 mod 18. Express your answer in terms of the…

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Q: (b) By applying part (a), solve the congruences 2x = 1 (mod 31), 6x = 5 (mod 11) 3x = 17 (mod 29).…

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Q: the congruence + 92 (2), using the method (Bertrand's Postulate). If n > 0 there erists a prime p…

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Q: 1. Solve the congruence x² + 1 = 0 mod 13³.

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Q: Solve the quadratic congruence x² = 35 (mod 817).

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Q: Find all solutions of x² = 27 (mod 253). Note that 253 is not a prime.

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Q: (a) Prove that 82n – 12" is divisible by 13 for all nɛ N.

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