Let n E N coprime with p. Show that there exists x, y E Z coprime and such that p | x² + ny² is and only if (=) = 1.

Q: Let {X;} be i.i.d.~ Exp(0) and a; > 0. Denote by Z = min{a,X1,..., amX,}. Suppose that 0. Prove that…

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Q: Given: LM = NO; LLMO = ZNOM Prove: ΔLΜΟ-ΔΝΟM M. N.

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Q: Let X = {x E Z|x = Y = {y € Z]y = 3n for some integer n}. 6m for some integer m} and let %D a. Prove…

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

A: Solution:     Let 3 does not divide n      => n = 3k ± 1 for some integer k.       => 2n2 + 1…

Q: Let p». Prove that, for every e > 0, a.a.s. the largest component of G(n, p) has size at least…

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Q: 0 <m < n such that am = e. (Hin

A: Subgroup generated by any element in a finite group will have at most n element where n is order of…

Q: B. Let H « G and K - G, and assume that HNK = {e}. Show that if x E H and y E K, then xy = yx

A: We can prove given result by definition of normal subgroup:

Q: Let 0 E A CR. Show that if A is both open and closed, then A = R. (Hint: Let M = sup{t: [0,t] C A})…

A: We prove this by contradiction.

Q: For any x ER, prove that there exists q E Z such that q-1<r <q.

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Q: Find c such that P(Z>c)= 0.3859

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Q: Prove that if Y:nz+P(x- mz) So thdz

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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: Given F(x, y, z) = (z, x, y) Solve for: а) divF b) curlF

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Q: 2, 5x .3 b) Y = In 60(х, + + X, 2. 3.

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Q: Let X and Y be defined as follows: X = {a E Z | a = 3x for some x E Z}, and Y = {b E Z |b = 3y – 3…

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Q: Let m eN and a, b e Z be such that (a, m) = 1 and (b, m) Let h = 0m(a), k = 0m(b), and let n = 0m…

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Q: Let P = {p:p=6x-1, for some x EZ} and Q= {q:6y+5, for some y EZ). Prove that P= Q

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Q: 7. (a) Find all the values of z such that e = 1+ i.

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Q: Assume that |Xn] 0. (Hint: first, prove that P(|X|>c+1) = 0.)

A: Answer:

Q: 2. Find +he source (メ) Q (x) that the So lutiom U (x it) of k + Qx) 2 - 0 l• Con e

A: given equation ∂u∂t=k∂2u∂x2+Qx,       -∞&lt;x&lt;∞           ...iux,0=0 claim- to find the source Qx…

Q: Find all the values of x in the interval [0,2pi], that satisfy the question.  10cos(x)+5sin(2x)=0

A: Given equation is Solve the given equation for x

Q: Let q E Q and x E R - Q. Prove that q + XER - Q.

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Q: Prove that for all values of the constant A, eAInt = e In.

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Q: Let p denote 3x E Z such that P(x)" and q denote dx E R such that P(x)". Ğive a simple definition…

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Q: 1. ( Prove that for any x, y e R, we have ||| - ||y|| < |× -y|l. 2. Let u e R. Prove that 0 u 0,…

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Q: Show that for every e>0 ther e exists P suuch that U (P, f) < U (f) + e

A: We will ise definition of U(f) and Infimum.

Q: If x > 0, prove that T - In (1+ x) > 2+2x

A: Using derivative

Q: Show that for every e > 0 there exists P suuch that U (P, f) < U (f) + e

A: Since U(f)=inf{U(P,f)}

Q: Prove: (i) b (A) CA if and only if A is closed. (ii) b(A) n A 9 if and only if A is open. (li1) b(A)…

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Q: Let w = z. In(xy² z®), then which of the following is- O a. O b. O d. O e. 0

A: While finding the partial derivative of a multivariable function, if we have to find the partial…

Q: Let a = 8 and b = 9. Find m,n E Z such that am+bn =1.

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Q: Find the smallest integer in the given set a.{ xϵ Z f x &gt; 0 and x = 4s + 6t for some s, t in Z }

A: Consider the equation: x=4s+6t……1   where: x, s, t∈ℤ and x&gt;0

Q: 30 Let A, B E M, and suppose that A² = 0 = B². Show that A and B are unitarily similar if and only…

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Q: Show that for any positive integers i and j with i > j, we have T;(x)T;(x) = [Ti+;(x) +T;-j(x)].

A: we have to show, for any positive integers i and j with i&gt;j TixTjx=12Ti+jx+Ti-jx Now, we know…

Q: Prove that there exist c1, c2 ER such that for all ne N C1 E1 sin (i)<c2 =D1

A: We will use the range of the sine function to determine the real constants c1 and c2

Q: EX: Find the Z. T. of X(X) = cas wok u(k) Soin

A: Z transform of the function

Q: If X-P(A) such that, P(X = 1) = P(X = 2) then find A. %3D %D O None O A=V2 O A=V3 O A = 2

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Q: 3. Consider tw o evénts A and B such that Pr(A)= 1/3 and Pr(B)= 1,2. Determine the value of Pr(B 0A)…

A: Given PrA=13 and PrB=12 and A and B are disjoint, that means PrA∩B=PrA×PrB…

Q: The characteristic of Zs[x]/ is 6. O None of the choices

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Q: Let P= {p:p=6x-1, for some x EZ} and Q= {q:q=6y+5, for some y €Z}. Prove that P= Q

A: Answer

Q: Find x, y, and z such that x³+y³+z³=k, for each k from 1 to 100.

A: By adding the cubes of positive integers we solve the given problem.

Q: Find the source Q cx) Such that the So lutiim Uckit) of + Q(x) 2 ə x² 0 < x <∞ | U(x,o) = 0

A: given equation ∂u∂t=k∂2u∂x2+Qx ,       -∞&lt;x&lt;∞            ...iux,0=0 claim- find the source Qx…

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

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Q: Prove that for every y E R, there exists a unique x E R such that 2(x – 1)3 – y = 0. %3D

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Q: Prove for a e Z and n e N that gcd(a, a + n) | n.

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Q: et A E C. Show that |A| 1 if and only if A= e for some 0 E R. %3D

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Q: 6. Let K be compact in R and let c E R. Prove that there exists a point a € K such that |c – a| =…

A: Given- Let K be compact in R and let c ∈ R.  To Prove- Prove that there exists a point a ∈ K such…

Q: Show that Oa = {a} if and only if a e Z(G).

A: For a∈G, Oa denote the orbit of 'a' under the conjuction action.The orbits under the conjucation…

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

A: to find the value of k

Q: Let P= (p:p=6x-1, for some x EZ) and Q = {q:q=6y+5, for some y EZ}. Prove that P=Q

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