4. Prove that an element r of Z2 \ 2Z2 is a perfect square if and only if x = 1 (mod 8).

Q: What is the number of solutions to the congruence in Z125? x³ + x² + 3 = 0 (mod 125)

A: As per the question we have to solve the following polynomial congruence equation : x3 + x2 + 3 ≡ 0…

Q: 2. Prove by mathematical induCHon that १) के) 22 32 42 2n is Hrue For all integers nz 2

A:

Q: 3. Prove that ox + 6x -4X12 is irreducible over Q.

A:

Q: Discrete

A:

Q: Show that x2 = Y mod p has

A:

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

A:

Q: Prove by mathematical induction that j=1 for all positive integersn. AI

A: To prove:                                                                        ∑j=1nj2≥∑j=1nj2,…

Q: 2=1 or 5 (mod 6), and if there exists a A-fold triple system ofv, v= 1,3 (mod6)| prove that

A: ANSWER:- Given that, λ=1 or 5(mod 6)   = 1 mod(6) or 5 mod(6) λ=1*6 or 5*6  =6 or 30

Q: Find the smallest positive integer x that solves the congruence: 7x≡5 (mod52)

A: 7x≡5 (mod52)---(1)

Q: x4 >is ........in Q[x] Not prime ideal ONone of the choices O Prime not maximal

A: Answer is maximal ideal , i.e , last option , and proof is given below.

Q: 3. Show that if n = 3 mod 4, then n cannot be the sum of squares of two integers.(i.e there are no…

A:

Q: Prove that for a positive integer n we have that 33n+3 – 26n – 27 is divisible by 169. - -

A: The given expression is: 33n+3-26n-27 This is to be proved that the expression is divisible by 169.

Q: 6"- (-4)" %3D k 1 k odd IM- 2]

A:

Q: 2. Prove that L(cashkt)= ki 52-K2

A: To prove that  L(cosh(kt))=ss2-k2

Q: Find the remainder using Fermat’s Little theorem 452 mod 11.

A:

Q: (5) Show that if n ɛ Z+ is odd which is divisible by k distinct primes, then the congruence x2 = 1…

A: Chinese Remainder Theorem: If a1,a2,...,ak are integers, and m1,m2,...,mk are pairwise relatively…

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: Prove that the number of solutions to the congruence xk ≡ 1 mod p is gcd(k, p − 1)

A:

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: b. Find the remainder using Fermat's Little theorem 452 mod 11.

A:

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

A:

Q: For p an odd prime in Z , Prove: If -1 is a perfect square in Z/pZ then p= 1(mod 4)

A: This can be done as follows:

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: Let be a rational prime, with p = 1(mod 4). Prove that p is the product of two Gaussian primes that…

A: We will use the basic knowledge of number theory to answer this question properly and completely.

Q: Prove that 67n+ 17 cannot be a perfect square for any three consecutive integers n.

A: We will use quadratic residue and proof by contradiction method.

Q: Verify that there is no magic square of order 2.

A: We will verify that there is no magic square of order 2 using the proof by contradiction.

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: 1 6 [: 2. Let %3D matnix c (by tnial and erron) Constauct a C (by tnial and eror using anly !,-1 and…

A:

Q: (a) Suppose that rị and r2 are quadratic non-residues modulo an odd prime p. Does x = r¡r2 (mod p)…

A: Quadratic Residues: Let m,p=1. m is said to be the quadratic residue mod p if the following equation…

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: Prove that ∀n ∈Z, gcd(7n + 1, 15n + 2) = 1 *use only the definition of divisibility and gcd please!

A: we want to prove that   gcd(7n+1 , 15n+2)=1

Q: Use the construction in the proof of the Chinese remainder theorem to find all solutions to the…

A: To solve the system of congruences: x ≡ 2( mod 3)  x ≡ 1( mod 4) x ≡ 3(mod5)

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: Number theory

A: Consider the provided question,

Q: Suppose that m, n are two integers such that m3 – 2n2 = 1. Does it follow that m and n are…

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: Use the construction in the proof of the Chinese remainder theorem to find all solutions to the…

A:

Q: 2. Find the largest negative x € Z that simultancously satisfies 4x = 9 (mod 5) 2x = 0 (mod 6) x =0…

A: Given : 4x = 9 (mod 5) 2x = 0 (mod 6) x = 0 (mod 4)

Q: (3) prove That () 3

A: We have to use z = x + iy.

Q: Use the Chinese remainder theorem to find all solutions to the system of congruences x = 1(mod 3),x…

A: Chinese Remainder Theorem- given n1,n2... where gcd(ni,nj)=1 , i≠j then the system of congurences…

Q: 3) If the gcd(z,n) = d and xz = yz mod n then: Show that x = y mod

A:

Q: Let x, y ∈Z. Prove that if x≡1(mod 5) and y ≡2(mod5 ), then x^2-y^2≡0(mod 5).  Give 2 examples to…

A: Given, x,y∈ℤ and x≡1mod 5, y≡2mod 5

Q: 2. Find the (general) solution of the following system by Chinese Remainder Theorem. 3r = 7 (mod 5)…

A: Given: The given values,

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: Suppose that m,n are two integers such that m3−2n2=1. Does it follow that m and n are relatively…

A:

Q: 2 There are integers m and n such that m² +n² is a perfect square.

A:

Q: Find the solution to 2x = 7(mod 17) that is positive and less than 17.

A: We find solution of given congruence relation