4) The least positive integer residue of 21000000 mod 17 is a) - 1 b) 1 c) 2 d) a

Q: Let n>,2 and k be any boretine Proce that integers (n-1)* | n, 2

A: Given, n≥2 and k is any integer. n≡1modn-1Therefore, ni≡1modn-1 ∀i∈ℕ…

Q: Assume that k mod 15 4. Compute 11k mod 15. Show all your work clearly.

A: Topic:- basic maths

Q: (4) Prove in two different ways that if a is a positive r number, then (a + 1)" > a" + na"-1:

A:

Q: Construct a proof using strong induction that any natural number greater or equal to 24 can be…

A:

Q: Use Fermat's little theorem to find the least residue of 8^70 modulo 23

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: 14. Demonstrate that 21 has no primitive root.

A: An integer n≥2 has a primitive root if and only if the the number n is one of the following; 2,4,…

Q: Solve the system of congruence's by using Chinese Remainder Theorem x =1 (mod 5) x =2 (mod 6) x = 3…

A: Consider the given information. x≡1 mod5x≡2 mod 6x≡3 mod 7 The general form of congruence is defined…

Q: Show that the set of a+bi : a€Z, b€Z, i=root of (-1) Gaussian integers has the same strength as the…

A: ANSWER :

Q: b. Prove that the integer 359,240,758 cannot be a perfect quartic. (i.e. fourth power) that 250 758…

A:

Q: Recall that tn denotes the nth triangular number. Prove the following identity: tn+2k = tn (mod k)

A: If tn is the nth triangular number, then tn=nn+12

Q: What is the smallest positive integer k such that xk = e for all x inU(7 . 17)? Generalize to U(pq)…

A:

Q: For which positive integers a is the congruence ax4 ≡ 2 (mod 13) solvable?

A:

Q: b. Prove that the integer 359,240,758 cannot be a perfect quartic. (i.e. fourth power) The 758 ppose

A:

Q: Consider 64z 16 mod 196. The number of solutions mod 196 is Oa None Ob 3 Oe 1

A: We find the number of solutions incongruent mod 196

Q: 5 is a quadratic non- residue mod 17 True False

A: 5 is a quadratic non- residue mod 17

Q: Let p be a prime and a a quadratic residue of p. Show that if p =1 (mod 4), ais also a quadratic…

A: Let P be an odd prime and Let a be a quadratic residue mod p ∵ P-a≡-a (mod p) ∴(P-a)p-12≡(-a)p-12…

Q: 2. Let a be a positive integer. Explain why the linear congruence az b (mod a + 1) has a solution in…

A:

Q: 1. Prove that 5 is irrational.

A: Consider the provided expression, 5 Show that the above expression is irrational. We proof by…

Q: . (a) How many quadratic residues are in the residue set Z19? (b) Determine if 4 and 9 are quadratic…

A: Introduction: Quadratic residue is a part of congruent modulo theory. It is a non-linear congruence.…

Q: 5. Use the method of successive squaring to compute 3875 (mod 103) 1

A:

Q: X=4(mod nod 13) X2-7(mod 15)

A:

Q: Suppose that p≥ 5 is prime. Show that -3 is a quadratic residue (mo p) if p = 1 or 7 (mod 12) and a…

A:

Q: Suppose that p is a prime with p = 5 mod 8 . Show that the congruence px2 + 6y = 1 mod 3p has no…

A: Consider the given information.

Q: (b) Prove by induction for all n >1 that an – bn = 2bn-1· %3D

A:

Q: 5. Explain why the congruence r = 2 (mod 4) has no solutions in Z.

A: We have to show that the congruence x2≡2(mod 4) has no solution in ℤ

Q: 23. Find de V100- tan O coton 2 Opbon 1

A: We have the given integral as ∫dx100-9x2 On solving the given integral furthermore, we get the…

Q: Find all the non-congruent solutions to  4x ≡ 6(mod 21)  express solutions using the stand…

A: Introduction: A linear congruence is generalized as ax≡bmod n. The greatest common divisor of a and…

Q: (B) Prove that (Q,|·|) is not complete.

A: A metric space (X,d) is said to be complete iff every Cauchy sequence in X converges to a point in…

Q: Given 51 distinct positive integers Strictly of Prove two trom to Sum 99 less than 100,

A: Please find the answer in next step

Q: Prove the following statement directly from the definitions. The product of any two even integers is…

A: The product of any two even integers is a multiple of 4.

Q: Suppose that p and q = 4p+ 25 are both prime. Show that p is a quadratic residue mod q.

A: Given p and q are both prime. and q=4p+25 To show that p is a quadratic residue mod q.

Q: Suppose that lal = 12. The number of distinct left cosets of in is: * %3D 12 O 10 O None of the…

A:

Q: 2. Prove by principle of Mathematical Induchion 6.7h - 2.3n is divisible by 4 for all nE 2#. 1

A:

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

A: To prove that quadratic residues (mod x) of all prime x contain y and y+2 where y∈ℤ Let us consider…

Q: . show that for every positive integer n, if n^2+2 is prime then n is a multuple of 3. Is the…

A:

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: prove that IIaind = nd(n)/2.

A: Analytic number theory

Q: Prove that if n, k are positive integers with k | (n +4) and k| (n – 1), then either k = 1 or k = 5.…

A: k | (n+4)  k | (n-1)  => k | {(n+4) - (n-1)} => k | 5

Q: 2. Prove that võ is iational number. 3. Ifa,beR. prove that ab

A: ANSWER 2:

Q: 10. Use Euler's Theorem to show that 5P(16), '=1 (mod 16). Show your work or explain your reasoning…

A:

Q: Solve the system of congruence's by using Chinese Remainder theorem x = 2 (mod 5 ) x = 3 (mod 7) x =…

A:

Q: 9. Use Euler's Theorem to show that 3P(26). =1 (mod 26). Show your work or explain your reasoning…

A: To show it by Euler's theorem :-

Q: Solve the linear congruence 5x = 3 (mod. 11). (Show all the work.)

A: The expression, a≡b (mod c)  implies that c divides a-b, that is when c divides a-b the remainder is…

Q: Prove that for n 2 2, F²r-1-F²n= Fn-1 Fn-2 Where F, denotes the nth Fibonacci number.

A: Facts:F0=0F1=1F2=1F3=2F4= F3+F2=2+1=3etc.Fn = Fn-1 +Fn-2

Q: Prove that tim 51

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Number theory

A: Assume a=x2

Q: 2. Show directly that sup(4) = 1 for A = {1- : n eN}.

A: As we know that, 1>1-1n ∀ n∈ℕ Therefore 1 is the upper bound of the set 1-1n:n∈ℕ. And for each…

Q: #2

A: