Suppose x F and you discover that x¹ = 1 mod p and x² = 1 mod p. Please prove that xgcd(m,n) = 1 mod p. Hint: Think about how you can apply Bézout's Lemma.

Q: Show that 4x2+6x+3 is a unit in Z8[x]. All that I know that I have to do is find a polynomial such…

A: Given polynomial is,

Q: Show that x2 = Y mod p has

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Q: Find a positive integer less than 6 for x that sat 2x + 340 = 1703 (mod 5).

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Q: Find all four solutions to x2 =6 (mod 19 x×23) with 1<x< 437.

A: Solution:-

Q: For integers a and b, prove that (a + b) 3 a3 + b3 (mod 3).

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Q: (b) Wilson's Theorem: Let p be prime. Then (p-1)!= -1(mod p). Hint: Verify for p = 2 and p = 3…

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Q: (2x + 3 ) ≡ 8 mod 12

A: (2x + 3 ) ≡ 8 mod 12 x + 3 + 9 ≡ 8 +9 mod 12  2x + 12  ≡ 12 +5  mod 12

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: Informal Exercise 20. Observe that 4| 102. Show that to find Rem(n, 4), you just need to replace n e…

A: We need to find the remainder when n is divided by 4 .

Q: (18)280 mod 125 =

A:

Q: x² + 4x = 10(mod11) (mod11) X=

A: Given that x2+4x≡10 (mod 11)x2+4x+4≡10+4 (mod 11)x2+4x+4≡14 (mod 11)x2+4x)(x+2)2≡14+11 (mod…

Q: #5. Suppose that a = 2 (mod 113), and b = 5 (mod 113) What is (a b + 56b + 155) mod 113?

A: given a≡2(mod113),b=5(mod113)a3≡23(mod113)a3b≡235(mod113)56b≡280(mod113)56b≡54(mod113)155≡15mod113)

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

A:

Q: Use Hensel's Lemma to find a solution to the following congruence: a +3 x + 3 = 0 mod 54.

A: Let f(x) be a polynomial , p is a prime number . If x=a is a solution of congruence equation…

Q: 2. Let f: Z-Z be given by the following rule: x+6 x=0 x = 1 x+8 x 2 f(x)=x+5 (mod 3) (mod 3) (mod 3)…

A: The given function is, fx=x+6x≡0 mod3x+5x≡1 mod3x+8x≡2 mod3 To Find: (a) Example to show that f is…

Q: (5a) Let A = {n E Z | n = 2 (mod 3)} and B = {n E Z | n = 1 (mod 2)}. Prove that if n E (A n B),…

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Q: (d) 357 862 193 = (mod 943). (e) 5327 6135 7139 2187 5219 . 1873 = (mod 8157). (Hint. After each…

A: ...

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: Exercise 6.3.4. Find all solutions modulo 10 to x² + 3x = 0 (mod 10). There are two distinct…

A: We employ the chinese remainder theorem to find the four solutions. Details typeset in latex in step…

Q: If a+c=b+c (mod m) then a =b (mod m) Select one: True False

A:

Q: Suppose that 3x ≡ 2 (mod 5). Find the remainder when x is divided by 5; in other words, compute x…

A: ANSWER :

Q: Find all integers sx such that 2x = 5 (mod 7).

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Q: Second Assignment: O Use Euler's Theorem to establish the following: 37 = a mod 1729. [Hint: 1729 =…

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Q: If a = 3 (mod12) then a does not equal 1 (mod6)

A: a=b mod n if and only if n divides a-b Now, we have that a=3mod 12 We need to show that a≠1mod 6 Let…

Q: Let f: N11 → N11 be defined by f(x) = (7x + 2) mod 11. Find f-¹ if it exists.

A: The first step is to check whether that the function f is bijective.

Q: Show that 2x = 1 (Mod 10) has NO solutions. Show work

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

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Q: If a (mod 0) = a, How would you interpret the congruence a ≡ b (mod 0)?

A: According to the given information, it is given that:

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

A: