Prove that (50!)2 mod 101 = -1 mod 101.

Q: Is there a solution to x^2 - 9x + 17 = 0 mod 73?

A: The given in this question is a quadratic equation in the form of x2-9x+17=0, and the objective is…

Q: Prove the following statement or disprove by giving a counterexample "for all positive intergers n…

A: We have to prove or disprove the following statement For all positive integer n and for all integer…

Q: 2n-1 b) Prove that if , r #1 (mod 2"), then n is even.

A: Given that x2n-1≢1mod2n . The objective is to show that n is even. The x2n-1≢1mod2n can be written…

Q: Show that 2 is a primitive root modulo 3" for all n > 1.

A: To prove that 2 is a primitive root modulo 3n for all n≥1.

Q: find x, where x= 9753 (mod 120) and -120<x<0

A: Ans: -87

Q: If x = y (mod n) and u = v (mod n) then which of the following is false. (a) x+ u = y +v (mod n) (b)…

A: Given that x≡ymod n and u≡v mod n. By definition, x≡ymod n implies x-y is divisible by n. Then let,…

Q: Prove that if ax = b (modn) then () x = () (mod ()) where d = gcd(a, n). Apply this result to 5x =…

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Q: If al = b (mod p), then prove that a = b (mod p).

A: (ii) given that                                                ap≡bp   (mod p) we have to prove…

Q: Compute the 3^201 mod 5.

A: Congruence value

Q: 3x = 1 (mod 6)

A: 3x≡1 (mod 6) ⇒ 3x-1≡0 (mod 6)This means 3x-1 is divisible by 6.

Q: Suppose x^3 = 1 (mod 41). Show that x = 1 (mod 41).

A: Given that: x3≡1mod 41. We have to show that x≡1mod 41.

Q: Determine the least residue of 56^1010 (mod 323).

A:

Q: prove that the following integer is composite (not prime). Do not use mod. 2^35(2^33-1) + 1

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Q: 2221 (mod 13) is equal to O 10 (mod 13) O 1 (mod 13) 7 (mod 13) 6 (mod 13) O None of these

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Q: If x = 2 (mod 5) and y = 4 (mod 5) then 2x – 3y =? (mod 5)

A:

Q: 2) Find 281171 + 171281 (mod 12)

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Q: (b) Find 95! (mod 96).

A: We have to find the value of 95!(mod 96). Since 95!=95*94*93*…*8*7*6*5*4*3*2*1.

Q: Solve : ex+ay =4 (mod 13) E 2 2 (mod 13) Using elimination

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Q: 8) Prove that 2 is a primitive root modulo 53.

A: We have to Prove that 2 is a primitive root modulo 53.

Q: (b) Compute 9-¹ (mod 19) by listing the multiples. Show your work.

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Q: Prove a2 = c² (mod n), assuming that a, c and n are integers with n > 1 and that a = c (mod n)

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Q: 6. Solve the equation x + 3x + 17 = 0 (mod 21)

A: The Chinese reminder theorem:                    Let p and q are coprime. Then the system equations…

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: Solve x21 ≡ 6 mod 7

A: If p is a prime number then xp-1≡1(mod p) for all x∈ℤ~pk  where pk denotes multiples of p . Suppose…

Q: The value of 52003 mod 7 is? (3 (4 8) 9

A:

Q: True or False : 4x ≡ 12 (mod 10) ⟹ x ≡ 3 (mod 10)

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Q: Prove by any method(contradiction/contrapositive or cases)   Prove that (a mod 2)(b mod 2)= (ab)…

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Q: *1. What is the least residue of 56 (mod 7) 58 (mod 7) 19458 (mod 7)?

A: Note that : Since you have posted multiple questions, we will provide the solution only to the first…

Q: (a) Prove that 318 = q (mod 19).

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Q: If x00 = 1(mod 109) then possible value of x is a) 108 (b) 218 (c) 109 (d) None of these

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Q: The remainder when aa is divided by 77 is 22 and the remainder when bb is divided by 77 is 55. (So a…

A: Given:          a≡2 mod(7)  and b≡5 mod(7)

Q: Demonstrate that the equation x10 + y10 - z10 = 5 has no integer solution. (work with modulo 11)

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Q: Compute (1001 mod d) for d = 2, 3,..., 10.

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Q: Let a € Z be odd. (a) Prove that a = 1 (mod 4) or a = 3 (mod 4). (b) Prove that a² = 1 (mod 4).

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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: Suppose that a and b are integers, a ≡ 7 (mod 19), and b ≡ 5 (mod 19), Find the integer c with 0 ≤ c…

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Q: Show that (-3) = 1 if q = ±1 mod 12 -1 if q = ±5 mod 12

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Q: Solve xS+x*+l =o (mod 34)

A: Given :                        x5 + x4 + 1 ≡ 0 (mod 34)             First, we solve it mod 3.…