For which primes p > 5 is -5 a square mod p? Your answer should be of the formp = {a1, a2, ..., ak} mod n , where a; € {1,., n – 1} for all i = 1,..., k. p = mod

Q: For which odd primes p is 3 a square mod p?

A:

Q: 3400 m is prime numbers, then compute 27 (mod m) 2. If m is natural numbers, show that a (mod m ) =…

A: Modular arithmetic: a≡b mod c means c divides a-b i.e., c|(a-b). So there exists a constant k such…

Q: 7) Let p be an odd prime, find if a) p =1 mod 8

A:

Q: Let a, be Z and n E N. For each statement decide whether it is necessarily true, or whether it can…

A:

Q: The integer n = 12700140x3243243 is missing the digit x. a) If n congruent to  3 (mod 9), then what…

A: According to the given information, Consider the given integer n:

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: Let p ≥ 2 be a prime. Suppose p does not divide a. Then ap−1 ≡ 1 (mod p).

A: The given problem is to show the given following modular congreunce that let p is a prime and p…

Q: Prove that for all integers x, if x is not divisible by 7, then x^3 ≡ 1 mod 7 or x^3 ≡ −1 mod 7.

A: Since you have asked multiple questions, we will solve the Question Q2 for you. If youwant any…

Q: Give an odd prime p congruent to ±1 modulo 12. Show that (3) = 1.

A: Please see the attachment

Q: an is an even number, show that a2 = 0 (mod 4), and if an is an odd number, show that a2 = 1 (mod…

A:

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

A:

Q: Take p and q are primes and p=3mod4 and q=3mod4. Take n=pq such that an integer z has a square root…

A: Given that р and q are primes and p=3mod4 and q=3mod4.

Q: Let p be a odd prime number and b, c € Z. Assume that there exists l e Z, such that l² = b² – 4c mod…

A:

Q: 5- If a = b mod (n), then ac = bc mod (n). 6- If a = b mod(n), then ak = bkmod(n) for every positive…

A:

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.

A: Consider the provided question,Hello. Since your question has multiple parts, we will solve first…

Q: Consider p> 2 a prime number, justify that: (P,') = (-1)" (mod p) to any 0 sn sp- 1,n EN. 1.1 1.2…

A: In the given problem we use the definition of combination. Then the problem is solved by…

Q: 4. Let p be an odd prime, and a, b primitive roots mod p. Show that ab is not a primitive root mod…

A: It is given that, p be an odd prime, and a and b primitive roots mod p.

Q: For which odd primes p is 5 a square modulo p?

A: As we know that  a is Quadratic residue mod p if and only if   a(p-1) /2 =1 mod p . If p and q are…

Q: Let n be an integer greater than 1. If (n-2)! ≡ 1 mod n, then Prove n is a prime number.

A: Use the Wilson theorem and different case to solve this problem as in next step.

Q: (a) Show that, for any odd prime p > 5, 1 if p = 1, –5 (mod 12) (주) -1 if p= -1,5 (mod 12)

A:

Q: For which positive values of k is nk = n (mod 7) for all integers n?

A:

Q: a. Let m be an integer such that m mod 5 = 2, what is 8m mod 5? b. Let n be an integer such that n…

A: Consider the given information: Let m be an integer such that m mod5 = 2. Then m=5k+2   where…

Q: 2

A:

Q: Let P be a prime, if ac = bc (mod p) and p does not divide c, then a = b (mod p) Select one: O True…

A: Given  gcd(c, p) = 1 and ac ≡ bc(mod p). (We may assume p > 1 so that c  ≠  0.) Then ac − bc =…

Q: Suppose that p is a prime number. Given this information prove that if x is an integer such that x^2…

A:

Q: (b) Proposition. For each integer m, 5 divides (m-m). Proof. Let m e Z. We will prove that 5 divides…

A: We need to find the correct proof of the given proposition.

Q: The last two digits of 52044 is 44. If ab = 0 (mod 3), then a = 0 (mod 3) or b = 0 (mod 3). F 10. F…

A:

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: For all integers a and b, if a = 7 ( mod 8)and b = 3 ( mod 8), then (a + b) = 2( mod 8).

A:

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: Let p be prime. (a) If a 1 (mod p), show that 1+ a + ...+ aP-1 = 1 (mod p). (b) If a -1 (mod p),…

A:

Q: If a has order n-1 mod n, then n is prime. O True O False

A:

Q: Consider modulo 11. a. Find 35 mod(11). b. Find a negative integer x such that x mod(11) is…

A:

Q: (c) For all integers a and b, if a · b = 1 (mod 6), then a =1 (mod 6) or b = 1 (mod 6). (d) For all…

A: (c) The statement is false. For example take a=5, b=5 ab=2525≡1 mod(6) Hence ab≡1 mod(6) But a≡5…

Q: Let p be an odd prime. Suppose p5 mod 8. Show p-1 p? – 1 mod 2. 5

A:

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: Let p > 3 be a prime. Show that if p2+ p + 1 is also prime, then p =5 mod 6.

A: We prove the result by the method of contradiction.

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: Let a,b,and c be natural numbers and n be a natural number > 0. Prove that if a = b mod n, then a^c…

A: Use mathematical induction to prove the given statement.

Q: Consider numbers of the form 9" mod 13, where n E Z+. We claim that all of these numbers will only…

A: First principle of mathematical induction : Suppose Pn is a statement where n∈ℤ+ . The statement is…

Q: show that If n is a positive integer and a is any integer prime to n, then a ln) = 1(mod n), where…

A: It's called Euler's generalization to Fermats theorem. I have used integer m instead of n.

Q: 29. How many different values are there for 4n(mod 7), as n ranges over all positive integers? 30.…

A:

Q: Write a formal proof for the following. For all integers a and b, if a ≡ 1 (mod 4) and b ≡ 3 (mod…

A:

Q: In the congruence equation 3x + 12 = mod 10, which of the following is FALSE O a. 5, 15, 25, 35,..…

A:

Q: Suppose n is a large odd number. You calculate 2(n-1)/2 = k (mod),  where k is some integer k is not…

A: Given that 2n-12≡kmodn Take the square on both sides, 2n-12≡kmodn2n-12·2≡k2modn2n-1≡k2modn (a) The…

Q: Let a, b, and c be integers and n a natural number. If a ≡ b mod n then a+c ≡ b+c mod n. How do I…

A:

Q: Q2/let n be a fixed positive integer and a,b be arbitrary integers. Then a = b (modn) iffa and b…

A: We have to prove that Given statement

Q: Using Wilson's theorem, to show that if p is a prime and p = 3 (mod 4), then ()! = +1 (mod p).

A:

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.

Q: For any integers m and n, if m mod 11 is 6 and n mod 11 is 7, what is m+n mod 11? 13 9 42

A: