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).

Q: _If A = Verify that [A-!]" = [A"]¬1 ? l1 2.

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Q: (1) F = B.C+A.C.D+A.C+E.B+E.(A+C).(A+D)

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Q: Find AB such that [3 1 21 T-1 A = 2 1 2 and B =|4 [1 2 2] -2 -5]

A: The solution is given as

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A: To find the value of a from the given summation.

Q: • G= *-3 ) • G = , det (G) = 10 %3D 4] Then x=12 O A) TRUE O B) FALSE

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Q: Show that x2 = Y mod p has

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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: Simplify the following expressions using Boolean theorems: F = B.C+A.C.D+ĀC+E.B+E.(A+C).Ã+D) (1)

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Q: If P(A) = P(B) = P(AB), show that P(AB + AB) = 0 [AB = An B]. %3D

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Q: show that |z3|  ≠ |z4|  it is verified that |z1+z2/z3+z4|  ≤ |z1|+|z2|/|z3|-|z4|

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Q: Prove that A2=I.

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A: Let A be a set and ∘ binary operation on A then 1. We say A is closed under the binary operation ∘…

Q: e. Then write the negation of the statement in Engli -ls for quantifiers. VmeZ) (m² is even). VxR)…

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Q: H 28. For all integers a, b, and c, if a b and a/c, then al(b + c).

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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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Q: C Compute the following expressions (d) Im a+bj a+cj Im [+

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Q: 3. Use congruences to prove the following: (a) 4"-1 is divisible by 3 for every natural number n.…

A: (a-b) is divisible by n.

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Q: (b) Let A and Bị be independent; similarly, let A and B2 be independent. Show that in this case, A…

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Q: 34 2(mod 8) is congruent modulo, then the equation of the congruent (a= b+km) O A. 34 = 2*4+26 %3D O…

A: If a≡b(mod m) then the equation of the congruent is a=b+km , where k is constant.

Q: If (ax + bx) = n and 7+1 E(ax – bx) - 2 Σ bk) , then ak 77+2 -1

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Q: Determine the value of k that satisfies the following identity (x is a variable). +k = In(x + ab?)…

A: Given equation is: lna+xb2+k=lnx+ab2 To evaluate value of k k=ln(x+ab2)-lna+xb2…

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A: (BA)T - ATBT + AC = AC In LHS, We know that, By the properties of Transpose, (BA)T = ATBT .…

Q: Select the description that characterizes the Boolean expression: xỹz CNF and DNF CNF, but not DNF…

A: Conjunctive Normal Form (CNF) is an expression which is intersections of unions. Disjunctive Normal…

Q: Let det(A) = 5 and det(A + B) = 20. We conclude that det(/+A¬}B) = 15 O True False

A: Given det(A) = 5  det(A+B) = 20  Now by multiplying det(A-1) on both sides  (det(A-1)) (det(A+B)) =…

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A: We have check given statement is true or not

Q: The first-order logic formula vx A(x)^3y -A(y) is satisfiable but not valid

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Q: (b) If r23= 1, prove that R² (23) = 112=113 and 01.23 = 01 (1-12).

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