Theorem 4.1. Assume (k, m) = 1. If {a₁,,am) is a complete (resp. reduced) residue sustem modulo 10 is ka

Q: Prove that if x,y> 0 then(x+y) ≥√√xy. For which x,y does equality hold?

A: Square of a number is always positive or equal to zero.

Q: 5. Suppose N > 1. (i). Prove that N is of the form 4n +1 if every prime factor of N is of the form…

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

Q: Section 1.8 Problem 5. Mark each statement True or False, and justify your answer. (T/F) A linear…

A:

Q: 4. Prove that for n > 0: E, 2* = 2"+1 – 1. sk3D0

A:

Q: 86. Show that Z, = r(cose + isine) and Z2 = r[cos(-6) + isin(-6)] are conjugates. %3!

A: Given,

Q: 49. Prove that x +> 2 (x1/2 – x-1/2)2. 2 for all x > 0. Hint: Consider

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Q: -Prove that a) b) n -("71) = (-+ 0 (," }) n r r+ n 2-0 (7) = 0 r=0

A: Also we know that    Binomial theorem

Q: a. [4]-[7] in Z17 b. [9][10] in Z12

A:

Q: 3. Let R-Z18. Determine the units and zero divisors of R giving the explicit unit or zero divisor…

A: We know that  no of units in ring Zn = phi(n), and  no of zero divisor of Zn = n - phi(n) -1

Q: 18.1.6. Is (x*) a prime ideal in Z[x]?

A: We have to check that whether  x3 is a prime ideal in ℤx  or not.

Q: 6.) Derike the binomial idendity nent) (un-) C Hint: For mzz, c^)+2(^) +m?]

A: To derive the binomial identity22+42+62+.........+2n2=nn+14n-16

Q: .55. (+) Let F be a field consisting of exactly three elements 0, 1, x. Prove that +x= 1 and that x…

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Q: Show that — ф+1 n=0 ||

A: Given φ is the golden ration satisfying 1φ=φ-1. To prove ∑n=0∞1φn=φ+1 It is known that φ=1+52.

Q: Let ECR. Prove that m*(E)= 0 implies that E is Lebesgue measurable.

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Q: 1. Lat A - [; : - and B = 2 2 1 3 4 (a) Compute AB. (b) Compute BA.

A: