2. Prove that for any m = 1 mod 11 that there are no integer solutions of m = 2r" - 9y.

Q: What is the number of solutions to the congruence in Z125? x³ + x² + 3 = 0 (mod 125)

A: As per the question we have to solve the following polynomial congruence equation : x3 + x2 + 3 ≡ 0…

Q: The linear congruence in two variables ax + by ≡ c (mod m), where a, b, c, and m are integers, m > 0…

A: As we know-the congruence ax+by≡c (mod m)has dm incongruent solution if   d=(a,b,c)|c& no…

Q: k) Prove that x = 5108 = 179 (mod 433) is a solution to the equation x² + 1 = 0 (mod 433), and use…

A: 52 ≡25(mod 433)54 ≡625≡192(mod 433)55 ≡960≡94(mod 433)56 ≡470≡37(mod 433)512 ≡1369≡70(mod 433)524…

Q: 6. Let p be an odd prime. Show that a2 = a (mod p*), k > 2 has exactly two solutions if x2 = a (mod…

A:

Q: If mx = 1 (mod n) has a solution, then nx = 1 (mod m) has no a solution. Select one: True False

A: We know the definition of Congruence ,  Suppose m be a positive integer , we say that integers a and…

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

A: To find:

Q: 2. Prove directly if the statements that are true, give counterexamples to disprove those that are…

A:

Q: 12. Prove that if gcd(a, n) { b, then ax = b (mod n) has no solutions.

A: To prove the given statement, it is equivalent to prove its contrapositive statement which is "if…

Q: 2. Let a be a positive integer. Explain why the linear congruence ax = b (mod a + 1) has a solution…

A:

Q: For which integers c, 0<c < 30, does the congruence 12x == c (mod 30) have solutions? When there are…

A: Given congruence is 12x≡c(mod 30). We know that ax ≡b(mod m) have solutions when gcd(a,m) divides b.…

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

A:

Q: 2. Prove directly if the statements that are true, give counterexamples to disprove those that are…

A:

Q: Find the four solutions of x* = 1 (mod 13).

A: Since you have asked multiple questions we will solve the first question for you .To get any…

Q: 3. Use the above theorem to prove that if p E N is a prime number such that p = 1 mod 4, then p is…

A:

Q: (5) Show that if n ɛ Z+ is odd which is divisible by k distinct primes, then the congruence x2 = 1…

A: Chinese Remainder Theorem: If a1,a2,...,ak are integers, and m1,m2,...,mk are pairwise relatively…

Q: Exercise 4.21. Prove that x2 = 1 (mod p) has exactly two solutions when p> 3 is a prime. 70

A: We shall solve it using basic knowledge from elementary number theory, in particular the properties…

Q: Prove that the number of solutions to the congruence xk ≡ 1 mod p is gcd(k, p − 1)

A:

Q: 13. a. Prove that for every integer n 1, 10" = (-1)" (mod 11). |

A:

Q: 4. The least positive residue of 2^1000000 mod 17 is: * 1 None of these

A: Question 01: By Division algorithm,  10,00,000= 16*62,500 and since 17 is prime and 17 is not…

Q: Let n ∈ Z. Prove that if n^2 + 2n −4 doesnt equal 1 (mod 5), then 5 doesnt divide n.

A:

Q: 13. a. Prove that for every integer n > 1, 10" = (-1)" (mod 11).

A:

Q: (a) Suppose that rị and r2 are quadratic non-residues modulo an odd prime p. Does x = r¡r2 (mod p)…

A: Quadratic Residues: Let m,p=1. m is said to be the quadratic residue mod p if the following equation…

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

A:

Q: 9. Show that if p is an odd prime and p = 3 mod 4 then p is not the sum of the squares of two…

A: In the given question we have to prove that if p is an odd prime and…

Q: Suppose that p and q = 4p+ 25 are both prime. Show that p is a quadratic residue mod q.

A: Given p and q are both prime. and q=4p+25 To show that p is a quadratic residue mod q.

Q: Find all values of m with 0 ≤ m ≤ 23 such that 10x ≡ m (mod 24) has a solution

A:

Q: Prove the following results. (I). If n = 2(mod 4), then n is the sum of three squares of integers.…

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Q: Find the first few terms of the sequence of pseudorandom numbers generated using the linear…

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Q: Number theory

A:

Q: 7. Let p and q be primes and suppose that p = 2q + 1. Let h E F and assume that h 1. Pre that ² h…

A:

Q: 2. Prove directly if the statements that are true, give counterexamples to disprove those that are…

A:

Q: Find an integer x satisfying the two congruences: x = 1(mod 3) 2x = 0(mod 6), or show that no such…

A: The given congruences are                                 x≡1 (mod 3)2x≡0 (mod 6)

Q: (c) Find all integers of order 6 mod 13 and all primitive roots mod 13. Show your working in both…

A: This is a question of Number Theory.

Q: Give all non-negative solutions for 10r =-17 mod 4.

A:

Q: If mx = 1 (mod n) has a solution, then nx= 1 (mod m) has a solution. Select one: O True False

A:

Q: Find all possible positive values of m: 47 = m mod (5)

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Q: Establish that if a is an odd integer, then 2" a“ = 1 (mod 2" *2) for any n 2 1.

A:

Q: Prove that for every odd prime 1000 < p < 2000 the number of –2, –1,2 which are squares mod p is not…

A: To prove that for every odd prime numbers strictly between 1000 and 2000 the numbers -2,-1, 2 which…

Q: Prove that for each integer a, if a not equal to 0 (mod 7), then a2 not equal to 0 (mod 7)

A: Consider the provided question,

Q: Show that for x and y integers, and m a positive integer x = y mod m implies x = y mod m.

A: This is a problem of Number Theory, Modular Arithmatic.

Q: Prove that there is no integer n such that n ≡ 4 (mod 10) and n ≡ 3 (mod 15)

A: We have to prove that there is no integer n such that n ≡ 4 (mod 10) and n ≡ 3 (mod 15)

Q: Does x = a (mod p) always have a solution for every value of a, whenever p is prime?

A:

Q: Find all solutions to the congruence 10x≡5 mod 15 (b) Prove that 7 divides 32n+1+ 2n+2for any…

A: In  (a) we use the extended Euclidean algorithm to solve the problem. In (b) we use the mathematical…

Q: 10. Find the sum of all incongruent solutions of a =1 (mod 17). (Enter the least non-negative…

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Q: e ab then th

A: Given: ak^2+bz^2 =c mod p To prove the given.

Q: Find the smallest positive integer xx that solves the congruence: 11x≡5(mod57)

A:

Q: 3. Let d = (a, n) where n > 1. Prove that if there is a solution to ax = b (mod n), then d divides…

A: