Suppose p is an odd prime and g and h are primitive roots of p. Show that for some odd integer k, h = gk (mod p).

Q: Find the smallest positive integer k such that 3k = 1 (mod 19).

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Q: Find an integer x satisfying the equation: 4x21 = 2(mod 5) or show that no such integer exists.

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Q: Find the smallest integer a > 1 such that a –1 = 2a (mod 11

A: To find:

Q: For each nonnegative integer i, what is the least residue modulo 9 or 10i?

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Q: Find all solutions in {0, 1,2, ..., 12} to x26 + 15x13 = 12 mod 13.

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Q: Find the primitive roots (mod 17).

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Q: Suppose that a is an odd integer and (a, 91) = 1. Prove that a12 ≡ 1 (mod 1456).

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Q: B- State Fermat little theorem, then find √√27 mod 31. Find (52525).

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Q: Find the primitive roots (mod 17).

A: Let p be a prime number and g be a positive integer. Then, g is a primitive root (mod p) if and…

Q: Suppose that p is a prime with p = 5 mod 8 . Show that the congruence px2 + 6y = 1 mod 3p has no…

A: Consider the given information.

Q: find the integer x such that 0 < x < 232 and 144x = 4(mod 233). Show your work.

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Q: a.) Determine if x< = 2 (mod 7817) has solutions. b.) Determine if x< = 3760 (mod 4861) has…

A: x2≡2(mod 7817) has a solution if and only if  2 is a quadratic residue of 7817. Euler's criterion…

Q: Let a and b be positive integers with (a, b) = 1. Show that if g is a primitive root (mod ab), then…

A: We have given (a,b)=1 where a, b both are positive integer. We have to show if g is primitive root…

Q: Suppose that a and b are integers with a = 11(mod 19) and b = 3(mod 19). Find the integer c such…

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Q: Find a number a such that 70a = 1 mod 17.

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Q: Find all integer values of x for which 3x = 4 (mod 8).

A: given congruence equation 3x≡4(mod 8)

Q: For any integer a, prove that a = 0 or 1 (mod 3).

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Q: (b) Find a prime p> 5 such that 2 and 3 are both primitive roots modulo P.

A: We test all the primes p&gt;5 and see which prime works.

Q: Prove: Let p be a prime number and m EN such that m <p². Then if p = 3 (mod 4) then the equation m =…

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Q: Suppose that a and b are integers, a = 4(mod 13), and b = 9(mod 13). Find the integer c with 0<c< 12…

A: We will use the basic properties of congruence modulo n relation to answer this question.

Q: Suppose that 3x ≡ 2 (mod 5). Find the remainder when x is divided by 5; in other words, compute x…

A: ANSWER :

Q: Find all integers sx such that 2x = 5 (mod 7).

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Q: List all the * = 10.

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Q: . Prove that for each natural number n, /3n +2 is not a natural number.

A: Let n be any natural number.  n=3m+k2   =3m2+6m·k+k2   =9m2+6mk+k2   =33m2+2mk+k2 Here, m and k are…

Q: Let p be a prime such that p = 1 (mod 4). Prove that -lF, is a square in Fp.

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Q: Let r be a primitive root of the prime p with p ≡ 1 (mod 4). Show that −r is also a primitive root…

A: The given problem is to show that -r is the primitive root of p when r is a primitive root of prime…

Q: Find, with proof, all primitive roots mod 27.

A: To find all primitive roots mod 27,

Q: Let p and q be two consecutive odd primes. Prove that p + q is divisible by at least 3 primes, not…

A: The details are given in step 2. The idea is to use a certain properties of primes.

Q: Find a number a between 0 and 86 such that 71122 = a (mod 87).

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Q: Find the smallest positive integer k such that 2k = 1 (mod 17).

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