Prove or give an counterexample to the statement that for any neZ+ and aeZn, the equation +n X = a has at most two solutions in Zn

Q: Show that 1 п-1 .. 1 X2 x5 п-1 x2 ... =II("; – 2;) i>j 1 n-1 .. •.

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Q: (a) For all a, b e Z prove that a³b² – ab* is even.

A: Properties of algebra Addition of two odd number is even  Addition of two even number is even…

Q: Find all values of z for which z* + 2 – i2v3 = 0 and represent them geometrically.

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Q: 1. Show that for all z1, Z2 E C, Then we have: |Z1 + Z2| < |Z1| + |Z2|

A: Using complex properties.

Q: Let a and b be positive integers. If ax + by = c has an integer solution (x, y), it is necessarily…

A: Consider the given equation. Find the characteristic for c if a, b are positive integers and ( x, y…

Q: Prove that |z₁+z₂²= |z₁|²+ z₂², if z₁/z2 is purely imaginary.

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Q: 3. Prove that {12a + 4b |a, b e Z} = {4c\c € Z}.

A: as per our guidelines we are supposed to answer?️ only one question. Kindly repost other question as…

Q: If n^2+6n+11 = x^2+y^2 for positive integer n, x and y prove [xy/2] is composite. Note: [t]…

A: Given :              n2+6n+11 = x2+y2              t ≤ t

Q: Q1(i) Prove that 1 1 1 + 2n²+1 V2n2+2 V2n2+n]

A: To prove: lim 12n2+1+12n2+2+.........+12n2+n=12

Q: Prove this! P (an) = W/ NM) [M-N (p-q)" ] 8 Npq %3D exp

A: your question is not clear i think some error in the question plz confirm it.....

Q: Let p be a prime. Solve the equation x2 = 1 in Z/p (i.e. find all solutions, and prove that there…

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Q: 7. (Hadamard) For any real nxn matria m 1/2 A)] < ΠΙΣa; and |det(A)| Σ i=1=j=1 holds iff either A…

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Q: Prove that E, k2* = (n – 1)2"+1 +2 for all n > 1. =D1

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Q: Prove that there are infinitely many solutions in positive integers x, y, and z to the equation x2 +…

A: We already know, (x,y,z) = (3,4,5) is a solution of the equation, x² + y² = z² i.e. 3²+4² = 5² Now,…

Q: 1. Given that r + px+q and 4x +q have a factor (x-r) where p,q and r are non-zero. Show that 4p…

A: topic - quadratic equations

Q: Show that for all z1, Z2 E C, Then we have: |Iz1| – |z2|| < |z1 +z2l

A: 2) The inequality can be proved as follows:

Q: Solve the following equations in Z6 with x expressed in both standard and non-standard names: (i) xx…

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Q: 30.Prove that if y1 and y2 have a common point of inflection t0 in I, then they cannot be a…

A: Suppose at the point t0, p and q is zero. That is, pt0=0 and qt0=0. Since t0 is a point of…

Q: 13 3. Use the IVT to prove that y = r Inr has at least one zero on 2 2

A: Above function is continuous function in given interval. So we can apply IVT

Q: Consider that: 1 + 2 + ... + n = An^2 + Bn + C  For all n and for some constant A, B and C.…

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Q: 1. Show that for all z1, Z2 E C, Then we have: |Z1+ Z2| < ]Z1] + |z2|

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Q: 1. Condense and expand. a) Condense as a single quantity: 1 In(3)+- In(x? +1)-3 In(1+x) 3 b) Expand:…

A: Given, (a)  To condense ln3 + 13ln(x2+1) -3ln(1+x) (b) To expand lnx2+1x+2(1+x)5

Q: 3. Let n E Z. Give a direct proof that there exist x, y E Z such that n = 55x + 78y.

A: Objective : To prove for any n∈ℤ ∃x,y∈ℤ such that n=55x+78y Prove for n=1, Greatest common divisor…

Q: Prove that if b+ 6 < 0, then x + bx + 5= 0 has at least one real solution.

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Q: Prove directly that no number of the form 4a (8b − 1) can be written as the sum of three squares by…

A: Let us first show that (8b-1) cannot be expressed as a sum of three squares Let us take any integer…

Q: Prove that Pn(1) = 1 and Pn (-1) = (-1)"

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Q: Solve the followig 13. BauSsian or Eduss Jordan Using E lemination. 5x, +6x=7X 3D0 3. ース×,+ 3X7…

A: Answer

Q: iii) a) Prove that Pa(-x) (-1)".Pn(x) u (2n)! 22 (n!) b) Prove that P2n(0) = (-1)".- %3D

A: To prove- Pn(-x)=(-1)n.Pn(x)P2n(o)=(-1)n.(2n)!22n(n!)2

Q: (1) Use the definition to prove that if xn → 2 then (2xn + 1) / xn → 5/2.

A: As per our company guideline we are supposed to answer only first three  qs, kindly post remaining…

Q: Prove indirectly: "V n E Z, 4 ł (n² – 2)."|

A: Ans:Let us assume that 4/n2-2;∀n∈zthen we have there exist a positive integer k such…

Q: (2) Determine k such kx? – 4x +÷= 0 that has real and different solutions. 2

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Q: Let S be the set of integer solutions to the equation x1 +x2+x3+x4 = 30, where x; > 0 for i = 1,2,…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Show that 1 is the only solution of x25 - 1 = 0 in Z37.

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Q: Show that the equation In(x + 1) = In(x²) has at least two real solutions.

A: we need to show that ln(x+1)=ln(x2) has at least  two real solutions.

Q: Find a number z that satisfies: |z| - z = 2 + i

A: Complex number which satisfy |z|-z= 2+i

Q: #8a using Descartes's rule of sign

A: Prove in the following that if all the coefficients of an equation are positive and the equation…

Q: Consider the following equations: 1= 1 2+3+4=1+8 5+6+7+8+9=8+27 10+11+12+13+14+15+16 27+ 64 %3D Form…

A: The sequence is given as, 1=12+3+4=1+85+6+7+8+9=8+2710+11+12+13+14+15+16=27+64 Now, the general…

Q: 6. Prove that, if a, b e Z, then 3a? – 4b – 5 0. - -

A: Answer

Q: Let r E Rt. Prove that if z – 2x2 +2 2021. -

A: Let x2=t, Now if x∈R+⇒t∈R+ ⇒t2-2t+2≤0⇒t-12+1≤0⇒t-12≤-1which is not possible for any value of t, for…

Q: If t,= 4n+1, find t,, t,, t3, and t Express t n+1, in its simplest form. n+1 In-1

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Q: Prove that there are no solutions in integers x and y to the equation 2x2 + 5y² = 14.

A: Two different proofs are given for the given problem as follows :

Q: 4. Prove that d x" = nx"-1 ミクー! de

A: To prove that ddx(xn) = nxn - 1

Q: Show that if A is singular and Ax = b, b = 0, has one solution, then it has infinitely many.

A: Given: A is singular and Ax = b, b = 0

Q: z, Z2 Prove

A: If z =x+ iy then  z = x- iy. i2 =-1.

Q: Find the value(s) of k such that A is singular.

A: to find the value of k

Q: (3b) Let n E Z. Prove that if 3|(2n² + 1), then 3 ł n

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Q: 4. Verify thatf (x) = x³ – x2 – 20x + 6 satisfies the three conditions of Rolle's Theorem on [0,5].…

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Q: 4. Show that: do Tb/(a? – b²) - a+bcos0 а b2 Where a>b>0 such that * -1+ 1. < 1 - a²

A: To Show: ∫0πdθa+bcosθ=πba2-b2 Concept: i. 1+tan2x=sec2xii. cos x=1-tan2x21+tan2x2iii.…

Q: 4. Prove that n(3n – 1)r x + 4x + 7x + ... + (3п - 2)x for n e N.

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