Let mi EZ for i=1,2,...,k and gcd (mi, mj) = 1 for all i ‡ j Prove that for x,y e Zif x = y (mod m₂) then, where m = m1 m2 mk. x=y (mod m)

Q: Find the first three values of the linear congruential generator X; = 2x;–1 +3 (mod 13) starting…

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Q: 4. Prove ve 2 b 2, 3/(a² +6²) 73/91316 there exists kez where xly means divides y x 1.e.

A: Let a or b be not divisible by 3. Then, we consider following cases.  Case 1: a is not divisible by…

Q: If x, y,z are integer's numbers and if x \ y then x\ z then: 1) Show that x\ ym - zn for m,ne z 2)…

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Q: Prove that Master Cheng has won precisely 21 times in a row during some consecutive days.

A: Please find the solution

Q: If x, y, z are integer's numbers then: 1) Show that if x \y then x\ yz 2) By use part 1 show that if…

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Q: 3- Suppose u, v EV and||u|| = ||v|| = 1 and = 1. Prove that u = v

A: Given: Vectors  such that ,  and .To prove that .Concepts used:Inner product space:Let be a vector…

Q: 3. If x is uncountable, then prove that A = {ECX: ECX: algebra O X: E is countable or E² is…

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Q: If S:= { / N - ¼/m : N,MEN, find wfs and sup 5

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Q: Suppose n E N and u, v € V₁ such that |u| = 5, |v|= 4 and u⚫v=-7 Find |3u + v. Find t R such that u…

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Q: If : = 4 cis e, find the modulus and the argument of iz

A: Formula used: The modulus of the complex number z=x+iy is given by z=x2+y and its argument is…

Q: Let A = and v= Verify A(u + v) = Au +Av. 2.

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Q: Suppose ū = 2j + k, ū = 2i – 4j + 2k and w = -5i – j+ k. %3D Compute the following values:

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Q: 14. Show that p q and (paq)V-PA) are equiv- alent.

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Q: Let A = -18 -15] 17 20 Find S, D, and S1 such that A = SDS-1 1 1 S- - BA] - PGI-1 D= = 1 2 0 -3

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Q: O show that 2. I| a+ V 1= 11 + 1i if and only if ū. ů =0 %3D

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Q: Verify the &-ô identity: Ejkɛipq = djpôkq – djqðkp. Then use this identity to show that (a) EilmƐjlm…

A: The given identity is εijk εipq=δjp δkq-δjq δkp  where the εijk is called the alternating tensor or…

Q: If x, y, z are integer's numbers and if x \y then x \ z then: 1) Show that x\ ym – zn for m,n e Z 2)…

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Q: Q₁: For (xy) e ZxZ, we say that x=ymodn if n/x-y. as Show that {(x, y) € Zx Z = x = y modo} = {(xy)…

A: (a) Let,           (a, b) ∈ {(x, y) ∈ Z×Z : x ≡ y mod 6}         => a ≡ b mod 6         => 6 |…

Q: Q6. Using the definition of x = y mod N, justify the rule x = x' mod N, y = y' mod N = x x x' = y x…

A: Using the definition of modulo N.

Q: 5. Let F= [3,2,1] and r= [2,1,–3]. Compute the following: %3D a) Proj „F. b) Orth rF = F – Proj „F.…

A: We know that  ProjrF=r.F∥r∥2r Now, r.F=23+12-31=5 ∥r∥2=r,r        =2,1,-3,2,1,-3        =22+11-3-3…

Q: Given that x>0 and y>0 proof that xy>0. (x,y are integers)

A: Given that x>0 and y>0, then to prove that xy>0. x, y are integers.

Q: Lot A and B be ovents such that P (A) = 4/5, P (B) = 1/3, P (AIB) = 1/6, where A' is the complement…

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Q: Prove ImA = A = Aln if A is m X n

A: Given: A be m*n matrix

Q: Q31 a) Prove that Re(iz) = -Im(Z)

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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: Let f₁=1,₂ = 1 and for all neN,n23,f=f+f2. Prove 1+√5 1-√√5 2 2 that for all neN, S √5

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Q: Let α belong to Sn. Prove that |α| divides n!

A: The symmetric group ​ is the group of permutations on n objects. Its order is .

Q: if zi = ayi for i=1,2,...,n, show that z=ay

A: Here we have given that if    then we have to show that   so here zi and yi are random variable…

Q: Define g: ZX Z→ Z, g (x, y) = z, where z mod 3 = x and z div 3 = y. Find g (2, 2).

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Q: The formula for slope between two points (x1, y1) and (x2, y2) is: m = 5-4 O True O False

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Q: Show that 2"=nCk and 0 = [(-1)^nck k=0 HINT: Consider(1-1)" and (1+1)" k=0

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Q: 2. Let f: Z-Z be given by the following rule: x+6 x=0 x = 1 x+8 x 2 f(x)=x+5 (mod 3) (mod 3) (mod 3)…

A: The given function is, fx=x+6x≡0 mod3x+5x≡1 mod3x+8x≡2 mod3 To Find: (a) Example to show that f is…

Q: Let A₁ = {1,2,3,..., i} fori = 1,2,3,... 72 Find Ai i=1 An A1 (0,00)

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Q: Show that gcd(a,n)=1 if these exist m and k such that ma+nk = 1.

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Q: 2 1] Assume A = 1 3 4. [2 2 4 and B = show that (AB)' = B' A' [3 6 4]

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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: Show that th equation x" + y² = z" (mod 5) has nonzero solutions in Zz if and only if n is odd.

A: Given: xn+y2n≡zn(mod 5)

Q: Consider the sequence defined by a_n= (4n + (-1)^n2 - 2)/8 for every integer n > 0. Find an…

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Q: Find a 3 x 3 matrix A such that A = 97 for all in R³. A =

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Q: xercise 4.2.9 Repeat the steps of the previous example where ad |) = 1/2, -42²) 3 1+2i -29] 1-2i -1…

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