Show that if x, y,z are positive integers then: z-gcd (x,y) = gcd(zx ,zy)

Q: Prove that det(kA) = kn det(A) for any A ϵ Mnxn(F ).

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Q: Show that For all z E C, with z # 0, then: |z/²z=1 = z

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Q: Suppose f [a, b] → R is integrable with the property that L(f, P₁) = L(f, P₂) for all pairs of…

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Q: (b) Prove that xe" 2 for some x in (0, 1). %3D

A: By the intermediate value property of continuous function if a function f:(a, b) --> R be a…

Q: Prove by mathematical induction: X n (U-, Y) = U=1(X nY), for V n E Z* – {1} and sets X,Y,,Y2,. Yp-…

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Q: Let x, y, z, w ∈ Z and n be a positive integer. Suppose that x ≡n y and z ≡n w. Show that x + z ≡n y…

A: Suppose that x ≡n y and z ≡n w then, (i) we are assumed that  x-y=k1 n  and  z-w=k2 n  for some…

Q: Functional theory and functional analysis

A: According to the given information it is required to prove that:

Q: prove that div A マ - Kr

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Q: Theorem .. ket a and b be non zerointegers- Then there exist Integers r and 8 Such that Hcf (9,b) =…

A: Theorem : Let a and b be nonzero integers. Then there exist integers r and s such that  HCF(a, b) =…

Q: Find all positive integers x, y, z satisfying x · y · z² = 5xyz. '*

A: Introduction :Given , positive integers x,y,zwe have asked to find the values of x,y,z which…

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

A: Let gcda,n=1.

Q: The number of reducible monic polynomials of degree 2 over Zz is

A: The number of reducible monic polynomials of degree 2 over Z3 is

Q: In Z[i], show that 3 is irreducible but 2 and 5 are not

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Q: Find the characteristic of Zm, × Zm, × · · × Zmr where m;'s are positive integers.

A: The ring ℤm1×ℤm2×...×ℤmk is a ring with unity 1,1,1,...k times Now, to find characteristic of…

Q: Compute In(z) dr.

A:

Q: Prove or disprove that 0 ≤ x2 - xy + y2 for all x and y in Z.

A: Consider the given inequality 0≤x2-xy+y2  for all x and y in  ℤ Now,…

Q: Show that 1 - i is an irreducible in Z[i].

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Q: Prove that ||A||1 = max1<j<n{£", |aij|}. %3D

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Q: (2) Suppose that G, C Zn-r. Then prove that G,-1C Zn-r+1

A: We prove that

Q: Show that x2 + x + 4 is irreducible over Z11.

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Q: prove for any 2 non negative integers x and y, where x>y, if their gcd IS Y, THEN THEIR LCM is x

A: The objective is to show that LCM(x,y)=x, given GCD(x,y)=y; where x and y are non-negative integers…

Q: Show that x2+x+4 is irreducible over Z11.

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Q: Show that if AT Ay = y for all y in Rn , then ATA = In .

A: Given that     ATAy=y  ∀y∈Rn. To prove ATA=In

Q: Prove that, IXI<a Rf and If -aL X Z a

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Q: Find all monic irreducible polynomials of degree 2 over Z

A: A polynomial of degree 2 in Z3 [x] is irreducible if and only if it has no roots in Z3.

Q: 10. Prove that Z has no smallest element.

A: we will prove it by contradiction, Suppose x belongs to Z  is the smallest element.

Q: Let Z ~ N(0,1). Prove for all z 2 0, P(Z > x) S

A: It is an important part of statistics. It is widely used.

Q: such that ab = 1 b) : In Z3XU3, Find the order of (1, 2). Show your work.

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Q: Let a in F_p, a is non zero then show

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Q: Let n EZ. Show that f(x)= x³ – nx +2 is irreducible over Q for n+-1,3,5. 72

A: Given: fx=x3-nx+2, n∈ℤ. To show fx is irreducible over ℚ for n≠-1,3,5.

Q: Show that if x is in both W and W-, then x = 0.

A: Dot product of perpendicular vectors is 0.

Q: In Let x, y ER" be given. (a) If ||x|| = ||y|| = 1 and x - y = 1, show that x = y. (b) If ||x|| ≤ 1…

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Q: prove that esch'x - In (* + x2 %3D

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Q: Prove that there are infinitely many triples (x, y, z) of positive integers satisfying x² + y? = z*…

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Q: Verify that the conclusion of Clairaut's theorem is valid, that is, that uxy = uyx u = x4y3 - y3…

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Q: #5 Let x, y, z be nonnegative real numbers such that y + z> 2. Prove that (x + y + z) >4x + 4yz.

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Q: Let m and n be positive integers with gcd(m,n) = 1. Prove that Z/mZ x _Z/ nZ = Z/mnZ

A: Suppose m and n are relatively prime.

Q: Assume that (8) ∞∑n=1nanxn−1=∞∑n=0anxn∑n=1∞nanxn−1=∑n=0∞anxn for all x, and determine what this…

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Q: Use Eisenstein Criterion to verify that , If p = 2, f(x) = x² – 2 is irreducible %3D over Q. State…

A: Explanation of the solution is given below...

Q: The union of two ideals is ideal* The only idempotent in an integral domain is 0 O T O F O O

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Q: Prove (x + y) + z = (x + z) + y without using induction.

A: It can be solve using group theory

Q: Prove

A: Let                 f(x)=x3+12x2+18x+6 By Eisenstein's Irreducible Criterion

Q: Show for all x, y in R |x+ys]xl + ]y!

A: consider left hand side  and use the result x=x2 x+y=x+y2=x2+y2+2xy≤x2+y2+2xy≤x+y2=x+y

Q: Prove that le-2z| 0.

A:

Q: z, Z2 Prove

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

Q: Show that if AT Ay = y for all y in Rn, then AT A = In.

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Q: Find all monic irreducible polynomials of degree 2 over Z3.

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Q: Let be an integral domain of characteristic 2. Show that Va,bED, (a+b)4=aª+b%.

A: Given that D is Integral Domain of characteristic 2. and a,b∈D  therefore  a+b∈D &amp; (a+b)2=e…

Q: Theorem. A rectangle R with side lengths 1 and x, where x is irra- tional, cannot be “tiled" by…

A: Without using the concept of linear algebra you can proof this. Let length of rectangle= 1 Unit…