Problem 1. (1) What are the squares modulo 4? (2) Let p be an odd prime and assume that p= a²+t for some a, be Z. Show that p=1 (mod 4). (3) Show that if p is an odd prime satisfying p = 3 (mod 4), then p is irreducible in Z[i).

Q: Za) Let A = {n E Z | n = 0 (mod 2)} and B = {n € Z | n = 0 (mod 4)}. Prove that if n E (A – B), then…

A: Follow the steps below.

Q: s 19 a prime in the Eisenstein integers? is 79? If they are, explain why, if not, display a…

A: Given: Primes in Eisenstein integer. To find: a) To check wheater 19 and 79 are primes in Eisenstein…

Q: Show that x2 = Y mod p has

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Q: Problem 2. Let n EZ be such that 3 n. Prove that n' =n (mod 63). (Hint: 63 = 7-9, (7, 9) = 1, and…

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Q: 1. Find a reduced residue system modulo each of the following integers. a) 6 b) 9 c) 10 d) 14 e) 16…

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Q: 1. Prove (2p+1) = 2 (mod p) when p is any prime.

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Q: Consider the three basic vectors: (2,1,0,-1),(-1,1,1,1),(2,7,4,1) a. give two vectors that are in…

A: Given the three basic vectors:                           (2,1,0,-1),(-1,1,1,1),and (2,7,4,1) We have…

Q: "EZ¹, 2n + 3 is prime."

A: We have to prove or disprove the given statement using given information.Concept required:1) One…

Q: (6) (a) = Let m be a positive integer and let a be an integer such that gcd(a, m) = 1. Prove that…

A: (a)Given that: m and a are positive integers and,.This implies that there exist integers u and v…

Q: 5. Let f(x) = 2x² + x + 4. It is known that f(1) = 0(mod 7). (i). Find a solution to f(x) = 0(mod…

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Q: Please show every steps. Do #2.

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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: Consider the primes p = 5 and q = 11. Let e = 3. (a) Compute pq and (p − 1)(q − 1). Prove that…

A: First, we need to compute  and . Given  and we can calculate  as follows: Now, we need to prove that…

Q: Consider the primes p = 13 and q = 19. Let e = 43. (a) Compute pq and (p − 1)(q − 1). Prove that…

A: To compute and .Given  and The value of and is calculated as follows: and .

Q: 1. Solve the congruence x² + 1 = 0 mod 13³.

A: To solve the congruence  , we can use the fact that  is a prime number.The congruence  implies that…

Q: 0001E JO du (u pou) 0=1-(4)"+ ... 4. Show that if a and m are positive integers with (a, m) = (a –…

A: Given a and m are positive integers with a,m=a-1,m=1

Q: In 1732 Euler wrote: "I derived [certain] results from the elegant theorem, of whose truth I am…

A: Details are provided in step 2.

Q: Compute the following (over Z2[x])(a) x3(x2 + x + 1) (mod x3 − 1).(b) (x2 + 1)2 (mod x3 − 1)

A: Step 1:To compute these expressions over Z2[x], we need to perform polynomial arithmetic under…

Q: 6. Let Fn 22" + 1, n > 1, be a prime Fermat number. Show that 3 is a primitive root modulo Fn.

A: Given: Fn=22n+1,n>1

Q: n). 10. Prove that a560 = 1 (mod 561) for all (a, 561) = 1.

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Q: Prove: Let be odd prime. Prove that Aor all i with 1 LikP-2), you have an +... + working so far:…

A: As per the question we just have to show that the prime p divides the following : xi + x2i + ... +…

Q: 2. Show that 1184 – 534 is divisible by 7. -

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Q: 1. Let a and b be natural numbers, where a, b > 2, such that gcd(a, b) = 1 and ab = (x + y)" for…

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Q: duce the expression for u to 2(5 ,Y = (1+5P) +4 (mod 2p - 1) or simply, to 2u, = 1+5P-1(mod 2p - 1).…

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Q: a₁xb₁ (mod m₁) a2xb₂ (mod m₂) ax=bk (mod mk.) has a unique solution modulo M = [1 mi, provided that…

A: This is called as the Chinese Remainder Theorem

Q: (7) Let m≥ 1 and let R= {0, 1, 2,...,m - 1}. (a) Prove that if r1, T2 € R and r₁ r2 (mod m), then r₁…

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Q: n 5.2.2 Prove by induction that for each natural number n, we have 2 = 2"+1 - 1.

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