Let c, d e Z and m E N. For each statement decide whether it is necessarily true, or whether it can be false. Justify your answer with a proof or provide a counterexample. (i) If c +d = 0 (mod m), then gcd(c, m) = gcd(d, m). (ii) If gcd(c, m) = gcd(d, m), then c+d=0 (mod m).

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A: (a) Here, the given statement is: For ∀n∈U=ℤ, p(n):n2+n is an even integer.

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: b=3(mod19)

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Q: If m ∠ D R C = ( 5 x + 42 ) ∘ and m ∠ C R A = ( 2 x + 75 ) ∘, then x =____?

A: Given,         ∠DRC=5x+42° and ∠CRA=2x+75°

Q: (6.1) Prove if r is a positive, even integer then 10* – 1 is divisible by 11. (6.2) Prove if r is a…

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

A: The given problem is to prove the following identities, we have to use basic identities of modular…

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Q: 2. |a + ib|= √a² + b² O True O false

A: This question is based on modulus of complex number.

Q: 3 S ₁² √ (²√₁² - ²√+²) ³ dt

A: Given

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Q: 10. Let a = (1,-1,2), b = (2,0,−1), and = (3,-2,0). Evaluate or explain why the expression is…

A: The vectors:

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Q: Let = [1, -1, 1], [-2, -1, 3] and = [-1, 2, 2]. Find (u ×7)•w: O-14 07 O 10 Does not exist

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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)

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A: Given: using contrapositive method

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A: "Since you have posted multiple questions, we will answer the first question. If you want the…

Q: Let A = and B = and [1 - [. C = -1 -3 -2 5 оооооооооо 38 -7 -2 -9 1 -1 -3 -8 0 -2 6-6 then which of…

A: Given:A=-1-3-25B=38-7-2C=-91-1-3-80-26-6A(rows=2,columns=2)B(rows=2,columns=2)C(rows=3,columns=3)The…

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A: We have check given statement is true or not

Q: A All of the above B(D-xXD+ x) = (D+xXD-x)

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A: Consider the given information. As we have given that a is divisible by b. then there is a number c…

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Q: 3. Let, a = 3, a2 = 4 and for n2 3, a, = 2an-1+ an-2 + n + 1, express a, in terms of n.

A: