Prove quadratic residues (mod x) of all prime x contain y & y+2 where y∈Z.

Q: 5. Let n Your-Reg-ID (7 digits number) and = let 21, 22, 23,..., Zn be the (distinct) n-th roots of…

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Q: show that for all integers x and y (x-y)3=x3-y3(mod3)

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Q: let x  be a fixed but arbitrary integer. Do there exist infinity many integer y such that,…

A: Let x = x0 be fixed given integer. Then,

Q: 7) Show that there are no primitive roots modulo 21 using EGLF and CRT.

A: Given value of n is 21 . Now, break 21 into its factor, 21=7×3 ,7,3=1 If m>2 and n>2 then the…

Q: k) Prove that x = 5108 = 179 (mod 433) is a solution to the equation x² + 1 = 0 (mod 433), and use…

A: 52 ≡25(mod 433)54 ≡625≡192(mod 433)55 ≡960≡94(mod 433)56 ≡470≡37(mod 433)512 ≡1369≡70(mod 433)524…

Q: Suppose that there exist integers x and y such that ax + by = 1. Show that a and b are relatively…

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Q: Let M = MT= -3 12 -6 15 Find formulas for the entries of M", where n is a positive integer.

A: we have the matrix M=-312-615 now we have to find the formulas for the entries of Mn where n is a…

Q: a + bi form modulus argument —3 — 3і - — 2 — Зі $1 · $2

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Q: Prove there are no natural numbers a and b such that √a+b = √a + √b.c

A: Assume such a,b exists and prove by contradiction

Q: If n = 12 m (me N), prove that "C₂ (2+√√3)² "Co- - + "C4 (2+√√3)4 "C6 (2+√3)6 ·+···=(−1)| 2√2 1+√3 n

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Q: Let u = (5,5,0) and v= = (5,5,5√√2). Compute |uxv|. Then sketch u, v, and uxv. |uxv= (Simplify your…

A: I have explained everything with in the solution. Please go through that.

Q: Tuesday In of (xin Exc (clip brow if flt) = √at +1, find "(2). (2 Wednesday

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Q: a) For an odd prime p, prove that the quadratic residues of p are congruent mod p to the integers…

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Q: nsist only of of even length. Generalize to find the EGF number of ations that consist only of…

A: To find: (1) The EGF for the number of permutations that consist only of cycles of even length.…

Q: 10. Let m and n are two positive integers with m > n. Let a be a positive integer. Prove that a2" +1…

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Q: 2. Prove that √2 & Q

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Q: 2 ľ xe2xdx

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Q: 3. Write the integral as the limit of a sum. 16 6 -Ź³ (5+46) (8) n-x n i=1 n B. Write lim3 5+: as a…

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Q: 1 Show that n2 + 1 IN V

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Q: 5. Prove that p(mk) = mk-1p(m) for all positive integers m and k.

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Q: Let A = 1 = {n+ (−1)n 2" | REN}. n Compute sup A and inf A if they exist.

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Q: prove f(x)=2x (mod 7) is injective

A: To prove that the function f(x) = 2x (mod 7) is injective, we need to show that for any two distinct…

Q: 4. Show that if x, y, z are integers such that x2 + y2 + z? = 2xyz, then x = y = z = 0. (H)

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Q: 16. Find the set of real values of x for which:/3 - 2x/S/x + 4/. N.Find the set of values of x for…

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Q: Prove this statement: we have numbers {a1, a2, a3 .,a2n} which are even and positive, but 0 < {a1,…

A: As, we know that, A.M.≥G.M. ---(1) A.M. of numbers, a1,a2,a3...a2n=a1+a2+a3+a4...+a2nn G.M. of…

Q: for ne J, x is a (1-x)(1+x+x²+.. + x^²) = 1=x^²+₁ variable, prove that

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Q: Find the first few terms of the sequence of pseudorandom numbers generated using the linear…

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Q: Find the first few terms of the sequence of pseudorandom numbers generated using the linear…

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Q: Queshon 6) Compute the arthogonal projection of 8 onto the Line L through -S] and the origin. -6 LS.

A: We have to find the orthogonal projection of v on the line through P and the origin.

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: ak + b) {{} = m – 1). k=0

A: Please cheek the details in the next step

Q: 1) i) Prove that if a and b are integers, with b>0, then there exist unique integers q and r…

A: A question in elementary number theory. We shall solve this using basic facts of divisibility.

Q: Find all integers sx such that 2x = 5 (mod 7).

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Q: M = 4 3 7 Find formulas for the entries of M", where n is a positive integer.

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Q: Let a1 = a2 = 1 and suppose an = an-1 + an-2 for n ≥ 3. Prove that a1a2 + a2a3 + · · · + a2n-1a2n =…

A: We have given that , a1 = a2 = 1 Also , an = an-1 + an-2 We need to prove that ,  a1a2 + a2a3 +…

Q: Prove that for а + —1,а > 0, пENis (1+ а)" > 1+ па.

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Q: 2- find aw dw if w = xy + ду +3/1

A: Given is the function w=xy+eyy2+1. We need to find ∂w∂y. In order to find the partial derivative…