The number of incongruent primitive roots modulo a prime p is: a) Ø(p) b) Фф — 1) с) р—1 O a O b

Q: Let x E Z and y E Z. Prove the following statement: If y = 2 mod 4, then xy 1 mod 4.

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Q: Given 6x + 2 ≡ 101 (mod 7), find an integer z such that 2x ≡ z (mod 7)

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Q: 25 Perform the mndi.cated -Computation Write the answer in scienh fic notation (4~10*/2.1

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Q: {X = I (mod 3) 2X= 2 (mod 5)

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Q: The square of an even integer is congruent to 0 (mod 4). Select one: O True O False

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

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

A: Please see the attachment

Q: Use the table in VII to give a formal proof of the following. For all integers a and b, if a = 1…

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

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Q: The elements of order 4 modulo 29 are O 6 and 12 O 7, 14, and 21 O 12 and 17 O None of the choices

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Q: Prove that if a and b are integers, then (a^5 + b^5) is congruent to a^5 + b^5 modulo 5.

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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: which of the below integers is congruent to 6 modulo 12 O 40 O 34 O 30 O none

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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…

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Q: Let m, n are integers. Prove that if m*n is even, then m is even or n is even. Specify what type of…

A: Given: m,n are integers. To prove: If m*n is even then m or n is even.

Q: How many integers from 1 to 1000: a) Are divisible by 7? b) Are divisible by 7 but not 11? e) Are…

A: a) From 1 to 1000 there are 1000 numbers. Now, 1000/7 = 142.86 So, there are 142 multiples between 1…

Q: 5. Let p = 61. Find all incongruent elements modulo 61, that have order: (а) 5 (b) 10 (с) 60

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Q: We'll now try to prove the following result in 2 steps: Theorem: Suppose that p is an odd prime. Let…

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Q: Suppose r is a primitive root mod a prime p such that p = 3 mod 4. Prove ord,(-r) =

A: Given: p≡3mod 4 To find: ordpp-g=p-12

Q: X=4(mod nod 13) X2-7(mod 15)

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Q: 5. Explain why the congruence r = 2 (mod 4) has no solutions in Z.

A: We have to show that the congruence x2≡2(mod 4) has no solution in ℤ

Q: A. If p 2q25 is a prime number, prove that p-q is divisible by 3.

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Q: The order of the integer 8 modulo 13 is: O 6 O 4 O None of the choices O 2

A: To find The order of the integer 8 modulo 13.

Q: Suppose p is an odd prime number, prove that p+1 (р — 3)! %3D (mod p) -

A: To prove: To prove that p-3!≡-p+12(mod p) for an odd prime number p.

Q: The elements of order 4 modulo 29 are O 6 and 12 O 7, 14 and 21 O 12 and 17 None of the choices

A: To find elements of order 4 modulo 29, we shall check each of these given options one by one. We…

Q: If p>=q>=5 and p and are both primes, prove that 24| p^2-q^2.

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Q: Given any odd prime p, prove that: 2(p−3)! ≡−1 mod p. (Hint: Start with 2(p−3)!≡x mod p. Then work…

A: Let p be any odd prime number.  So we start with  2p-3!≡x mod p Also we have p-2≡-2 mod pp-1≡-1 mod…

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: In the formula a b (mod n), what is the value of b in 36 b (mod 5)?

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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: Find all prie 8. If p > q 2 5 and p and q are both primes, prove that 24 p2-q?.

A: Given: p≥q≥5 and p and q are both primes

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

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Q: Explain why the congruence r2 = 2 (mod 4) has no solutions in Z

A: Result: Square of all integers modulo 4 is either 0 or 1.…

Q: Use Fermat's theorem, show that if ged(a, 155) 1 then a1 = a (mod 15

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Q: Let p, q, and r be primes other than 3. Show that 3 divides p2 +q2 + r2.

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Q: Assume a,b,p € Z and p is prime. Prove that if plab, then pla or p b.

A: We have to solve given problem:

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

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Q: the pth term of AP is a and the qth term is p,prove that it's nth term is (p+q-n)

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Q: If g is a primitive root modulo 23, then for which 1 si< 23 is g' also a ive root?

A: Answer:

Q: Two integers are congruent modulo to a)prime integer b)odd integer C) not a prime nor composite…

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Q: 4. Given the Ecryption code R = f(S) = (3S+5) (mod 26), find the decryption code ( i.e. %3D find…

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Q: Let p and q be prime numbers. (1) If p is not equal to q, how many generators does Zpq have? (2) If…

A: Given p and q be primes.

Q: Given that a, b, c,m are integers with m > 0. If a = b (mod m) then prove that ас3 bе (mod m)

A: we have to use the only definition of congruence and the definition of divisibility

Q: Let p, q, and r be primes other than 3. Show that 3 divides p2 + q2 + r2.

A: We have to prove that 3 divides p2+q2+r2, where p,q and r primes other than 3.

Q: Prove that If p is prime, p divides k^2 + m^2 and p divides m^2 + x^2, then p divides k^2 + x^2

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Q: Prove the following statements using either direct or contrapositive proof. 1. Let x,y eZ. If x+y is…

A: Please see the below picture for detailed solution.

Q: 2x = 1(mod 5) 31( mod 7) 5x 9(mod l)

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Q: (b) Solve the set of congruences 2x =1 (mod 5) x=3 (mod 4)

A: Note:- We’ll answer the first question according our protocol. Please submit a new question…

Q: is a prime number and that n is an integer such that p|n^2 prove that it follows p|n

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