4. Let a, b, c, d, n € Z such that a = b (mod n) and c= d (mod n). (a) Prove that a+c=b+d (mod n). (b) Prove that ac = bd (mod n). (c) Prove that am = fm (mod n) for all m € Z>0.

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: Let p = -1 mod 3 be prime. (a) Show that there exists d with 3d = 1 mod p – 1. (b) Show that if a³ =…

A: As p≡-1 mod 3 Let q∈Z ,such that p=3q-1 p-1=3q+1 ⇒3(p-1)⇒gcd(p-1,3)=-1 ⇒3∈u(p-1) where, u(p-1) ={x…

Q: Problem MCD = Greatest common divisor (GCD) t them.a, b, c and m €Zsuch that m 22. a-). Prove that a…

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Q: If a cy (mod n) and c> 0 then a) x y (mod en) b) x y (mod n) ) x=y (mod) d) None of these :Select…

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Q: 1. Let a, b, c, r, s, and t be integers with r>s>t>1. If a = b (mod r), a = c (mod s), and gcd(r,s)…

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Q: If a = b (mod n) and a = c (mod n) then which of the following is true. (a) b3 c (тod n) (b) b2 =…

A: Given below the detailed solution

Q: 2. If a = 3 (mod 11) and b = 9 (mod 11), compute the values, positive and fully-reduced modulo the…

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Q: [a, b] = N=1 (a -,b+-) To prove this, show the following: То Show that [a, b](a – ÷,b +- ) for every…

A: Given that, [a,b]=∩n=1∞a-1n,b+1n This can be written as, ∩n=1∞a-1n,b+1n=[a-1,b] The aim is to show…

Q: 1. Find all x, y, and z such that a) x = 3 (mod 4) x = 5 (mod 21) x = 7 (mod 25) b) 3y = 9 (mod 12)…

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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: Let p be prime. Suppose a, b ∈ ℤ and ab congruent to 0 (mod p). Prove that either a congruent to 0…

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Q: Prove or disprove by using "congruence modulo n" and "divides." (a) For all positive integers a, x…

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Q: Let n be a fixed positive integer greater than 1. If a mod n = a' andb mod n = b', prove that (a +…

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

A: Given that x≡y (mod n) and u≡v (mod n)

Q: Let a, b, c and meZ, and m 2 2 1. Prove: if a =b (mod m), then GCF (a, m) = GCF (b, m) %3D m 2.…

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Q: Suppose x F and you discover that x¹ = 1 mod p and x² = 1 mod p. Please prove that xgcd(m,n) = 1 mod…

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Q: Let p and q be two distinct primes. (a) Prove that if a = b (mod p) and a = b (mod q), then a = b…

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Q: Let (a,b)=1 with a = c (mod b) and bª = c (mod a) for some integer c. Prove that ab+(b). +ba+(a) = c…

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Q: a) Show that 2340 ≡ 1 (mod 11) by Fermat’s little theorem and noting that 2340 = (210)34.b) Showthat…

A: According to the given information, Fermat’s little theorem says that:

Q: 3. Let a, b € Z. (a) Prove that (a + b) = a + b (b) Prove that (a + b) a +b mod 5. mod 4. Use the…

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Q: Prove that if a = b (mod m) and if c = d (mod m), then ac = bd (mod m).

A: a≡b   (mod m)⇒ m|(a−b)That is there exist an integer k such that a−b=mk

Q: 10. We say that an integer a is nilpotent modulo m if there exists a pos- itive integer k such that…

A: To prove that an integer "a" is nilpotent modulo "m" if and only if "a ≡ 0 (mod rad(m))," we need to…

Q: Recall that a = b( mod n) if a = b+ kn for some integer k. Use this to prove that if a = b ( mod n)…

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Q: Prove that if x2 is congruent to y2 (mod n) and x is not congruent +or - y (mod n)  then gcd(x + y,…

A: Given: x2≡y2modn x≢±ymodn

Q: Suppose that a and b are integers with a = 11(mod 19) and b = 3(mod 19). Find the integer c such…

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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: Prove or disprove by using "congruence modulo n" and "divides." (a) For all positive integers a, x…

A: (a) For all positive integers a, x and y, if (a + x) ≡ (a + y) (mod 12), then x ≡ y (mod 12). (b)…

Q: (9) Let a, b, and m be integers with m 21 and (a,m) = 1. Prove that the congruence ax=b (mod m) has…

A: a, b and m are integers with m ≥ 1 and (a, m) = 1.We will prove that ax ≡ b  (mod m) has a unique…

Q: 3. For the following statements, either prove that they are true or provide a counterexample: (a)…

A: Given that, Let a, b, m, n ∈ Z such that m, n > 1 and n | m. a ≡ b (mod m) By using the…

Q: 16. Prove that an integer (a,-1an-2 · . . ao)10 is divisible by 11 if and only if ao + a2 + a4 + . .…

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Q: a) Find a0, a1, a2, a3 so that 1234 = a3 · 10^3 + a2 · 10^2 + a1 · 10^1 + a0 · 10^0 b) Prove that…

A: NOTE: According to guideline answer of first question can be given, for other please ask in 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: Suppose gcd(a, n) = 1. Then a(") = 1 (n

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Q: Let 1 ≤k≤p-1 where p is a prime number. Then prove that (₁7¹) = (-1) (modp) k

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Q: Suppose that a1 ≡ b1 mod (m) and a2 ≡ b2 mod (m) -Show that a k 1 ≡ b k 1 mod (m) (hint: use…

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