Show that if p is a prime number then every equation of the form ax ≡ b mod p with a 6≡ 0 mod p has at most one solution. Give examples to show that linear equations mod n (for n not prime) that can have zero or multiple solutions.

Q: Show that if a and b are integers and m and n are positive integers such that n|m, then If a = b(mod…

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

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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: 2. Give an example to show that a? = b2 (mod n) need not imply that a =b (mod n). 3. If

A: Given below the detailed solution

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

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Q: 2. If a = 1 (mod p), then a=1 (mod p). Select one: True False

A: If , then . The truthfulness of the given statement can be proved by taking an arbitrary value for…

Q: Exercise 37 Find an n where x² = -1 (mod n) has a solution and an n where x² = –1 (mod n) has no…

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Q: 24. Prove that for all integers m and n, if m mod 5 = 2 and n mod 5 = 1 then mn mod 5 = 2.

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Q: Show that: Ifp is odd prime of the form 4k+1, then find a solution of x +1 0 mod p

A: We shall use Euler’s Criterion here. Euler’s criterion states as follows: “Let p be a prime. The…

Q: Let n and a be positive integers and let d = gcd(a, n). Show that the equation ax mod n = 1 has a…

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Q: For which primes p > 5 is -5 a square mod p? Your answer should be of the formp = {a1, a2, ..., ak}…

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Q: For which positive integers a is the congruence ax4 ≡ 2 (mod 13) solvable?

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Q: Consider 64z 16 mod 196. The number of solutions mod 196 is Oa None Ob 3 Oe 1

A: We find the number of solutions incongruent mod 196

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

A: Concept: The quadratic congruence: x2≡a(mod p) , where gcd(a,p)=1 Has a solution if (ap)=1 That is a…

Q: a) How can an inverse of a modulo m be used to solve the congruence ax ≡ b (mod m) when gcd(a,m) =…

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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: 13. (a) Prove that for each integer a, if a 0 (mod 7), then a #0 (mod 7).

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Q: Show that if a ≡ b (mod m) and c ≡ d (mod m), then a + c ≡ b + d (mod m).

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Q: 2. Give an example to show that a (mod n). b (mod n) need not imply that a =b

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Q: 4. Exercise $2.5 #58a. Let a and b be integers not divisible by the prime mumber p. (a) If aP = mod…

A: Given the data

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: 4. (a) Solve the set of congruences х 1(mod 3), x= 2(mod 4), x = 3(mod 5).

A: We  will solve it by Chinese Remainder Theorem Given  x≡1(mod3)x≡2(mod4)x≡3(mod5) Compare it with…

Q: Recall that a = b( mod n) if a = b+ kn for some integer k. Use this to prove that if a = b ( mod n)…

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Q: Show that if p and q are distinct primes, then pª-1 + q®=1 = 1 (mod pq).

A: Given problem is to show the following identity using congruence, we have to consider that…

Q: show that 2 is a primitive root mod 3^e for all e>=1

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Q: Prove that for any prime, p = 1 (mod 4) there are exactly (p-1)/4 incongruent values of a 0 (mod p)…

A: By theorem (a,p)=1 then xn≡a modp has g.c.d(n,p-1) solution or no solution according as…

Q: Let m1, m2,…, mn be pairwise relatively prime integers greater than or equal to 2. Show that if a ≡…

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Q: (b) Prove that if a = 7 mod 10 and b = 6 mod 15, then a + b = 3 mod 5.

A: Our objective is to prove . Since you have posted multiple questions I am solving one.

Q: (4) Let p be a prime number. Then r = 1 mod p has exactly two solutions modulo p.

A: 4) To prove, x2≡1mod p has exactly 2 roots such that p is a prime number.

Q: Prove that if a1, a2, . . a, are n > 2 integers such that a; = 1 (mod 3) for every integer i (1 < i<…

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

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Q: Exercise 37 Find an n where x? = -1 (mod n) has a solution and an n where x2 = -1 (mod n) has no…

A: Take n=5 and n=4

Q: You and your friend Lara are considering the linear congruence ax = b (mod m), where m = 6 x 11 x…

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Q: Show that: if a = b(mod n) and C = d\modn), then a +c =b+d\mod n)!

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Q: which one will give a solution of [6,13] a)4x is identical to 3 mod 7 b)9x is identical to 11 mod…

A: which one will give a solution of [6,13]

Q: (d) For all integers a and b, if ab = 7 (mod 12), then either a = 1 (mod 12) or a = 7 (mod 12).

A: (d) For all integers a and b, if ab=7 mod 12, then either a =1 mod 12 or a =7 mod 12

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

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

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Q: Show that if p is an odd prime and a is an integer not divisible by p, then the congruence x2 ≡ a…

A: According to the given information, Let p is an odd prime and a is an integer not divisible by p…

Q: Exercise 36 Prove that x? = a (mod p) has either no solutions or 2 solutions when p is a prime and a…

A: I have use the method of contradiction

Q: Find a two digit integer c such that 101c = 1 mod 107. This congruence has a unique solution mod 101…

A: We need to find a digit integer c such that 101c ≡ 1 mod 107. As per the given question, this…

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: Which are true for modulo 12? 1.) Does 5x ≡ 6 mod 12 have a solution? How many? 2.) Does 4x ≡ 6 mod…

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Q: If ab = 0 (mod m), then either a = 0 (mod m) or D=0 (mod m). Select one: O True False

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Q: 3. Find all integer solutions to the pair of congruences (if any) x = 1 (mod 2) x = 2 (mod3) x = 3…

A: To Find: Integer solutions to the pair of congruences(if any) x≡1(mod 2), x≡2(mod 3), x≡3(mod 5)…

Q: For which primes p > 7 is 7 a square mod p? Your answer should be of the form p = {a1, a2,... , a}…

A: Given: p&gt;7 ,If 7 is a square mod p.