5. Let a, b e Z and n E Z*. Prove that if a = b(mod n), then gcd(a, n) = gcd(b, n). (Hint: use the fact that if an integer d divides two numbers, then d divides any integer combination of those two numbers.)

Q: Let n E Z+, n > 2. Show that n is prime if and only if (n – 2)! = 1 (mod n).

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Q: Show that if a and b are integers and m and n are positive integers such that n|m, then If a = b(mod…

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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: T F 5. To prove the statement: If ab =0 then a = 0 or b = 0, you may assume ab = 0 and a # 0 and…

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Q: Let a, be Z and n E N. For each statement decide whether it is necessarily true, or whether it can…

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Q: Question 2 Let a, me Zt and (a,m) = (a-1,m) = 1. Then prove that 1+a+...+am)! = 0(modm) E

A: Eulers theorem will be used to solve the problem.

Q: 1. Show that k1601 = k (mod 17) for any k.

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Q: et x, y E Z. We have (x+y)³ = (x+y)² (x+y) = (x² +2xy+y?)(x+y) = (x³+3x?y+3xy² + y³) = x³ + y3 od 3.

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

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

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Q: y 10 8 6 4 2 X -5 5 -24 Amazing work. Find the domain of the function. (Enter your answer using…

A: Solution :  Domain of the function is  (- ♾️ , 3] union [3, ♾️)

Q: 1. Let a, b e Z. Prove that if a = 0 (mod 5) and b = 2 (mod 5), then a² + b² = 4 (mod 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: Let n e Z. Prove that n² + 4 is divisible by 8 if and only if 3n + 6 is divisible by 12. (Hint: the…

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Q: 15 Composite Then and O modn Show that if n there are with X#0 od n and y7 but where yy= Omod n.

A: Given n is a composite number

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 n E Z. Prove that n2 # 2 (mod 3).

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Q: Let n be a positive integer and let ged(b, n) = 1. Show that bx by (mod n) if and only if x=y(mod…

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Q: et r +1 be a real number. Use induction to show that for any positive integer n, a(rn+1 – 1) a + ar…

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Q: Prove that if a = b (mod m) and if 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: 3 Let A =2E -1 0 B = 0 2 If BXT = A, then X= Select one: -3 0 a. 1 -3 O . 1 -3 O d. None of these 1)

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Q: 2. Prove that if ar b (mod n), then () x = () (mod ()) where d = gcd(a, n). Apply this result to 5x…

A: We first find the gcd of the given numbers and then use that to solve the given congruence. The…

Q: (12) Let a,b, m, nE Z. Assune if greater than ave Prove mln and a = b (mod n) that then a = b (mod…

A: Congruence equations

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

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Q: 2. Let A = {n? +n € Z:n e Z} and B = {r e Z : r³ – 3x³ + 2x = %3D O mod 5}. Prove or disprove: A C…

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Q: Find d

A: The given question involves the concept of modular arithmetic and the use of the modular inverse.…

Q: - Let a, b, n be positive integers. Prove that if a = b (mod n), then (a, n) = (b, n).

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Q: For each pair of integers m and d, calculate m mod d. Show your work! a. m = 43 and d = 9 b. m 50…

A: If m and d be two integers . Then m mod d means that the remainder when m is divided by d .

Q: Given the expressions z=3 +i and w=31.Find modulus of zw and modulus| and argument of W and argument

A: In this question, we need to find modulus and argument of the given numbers.

Q: 6. Prove that if gcd(a, b) 1, then gcd(a +b, ab) = 1. %3D %3D

A: Given that gcd(a,b)=1 If possible , let us assume that  gcd(a+b,ab)=d,d≠1 Then the number d must…

Q: 7. Let n E Z. Prove that 9n3 6 (mod 27).

A: The objective is to prove the above statement.

Q: Exercise 5.2.4. Let n, q € N, and let a, b = Z. Suppose that a = b (mod n), and that q|n. Prove that…

A: Our objective is given below:

Q: If a =b (mod m) then a" = b" (modm) Vn e Z*.

A: Given a≡bmod m. we have to show that an≡bnmod m. we wil prove this by result by using method of…

Q: T T T F F F 11. 12. 13. If ab = 0 (mod 3), then a = 0 (mod 3) or b = 0 (mod 3). If ab = 0 (mod 6),…

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