Using the Law of Quadratic Reciprocity, prove that for an odd prime p ≠ 3 (3/p) = { 1 if p congruent to +- 1 (mod12); -1 if p congruent to +-5 (mod 12). Show 4 cases.

Q: Use a direct proof to show that the sum of two odd integers is even.

A: Types of integers : Odd integers : The integer which is not divided by 2 . It has the form 2k+1…

Q: Find all primes that are the difference of the fourth powers of two integers.

A: Let the two integers are x and y and n is prime so we have  n=(x4-y4)       (suppose x>y)…

Q: Use the Chinese Remainder Theorem to find a solution to the congruences ´x = 2 mod 10 x = 3 mod 1001…

A: The given system is, x≡2mod10x≡3mod1001x≡3mod9 Chinese reminder theorem: If m1,m2,…,mn are positive…

Q: Prove by induction that 7 divides 3^(5n+3) + 5^n for every non-negative integer n.

A: We can prove it by using mathematical induction.

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

A: Let, the system of congruence is: x≡a1mod n1x≡a2mod n2x≡a3mod n3 Then, there exists integers…

Q: he sum of any two integers of the from 4k + 1isalways: COdd prime Zero even

A: The sum of two consecutive integers can always be written as 2n+1, but it can be written as 4n+1…

Q: Prove that if n is an integer and 7n+2 is even, then n is even. State which method you are using for…

A: Let  n be any integer.  We have given that , 7n + 2 is even. We need to prove that ,  n is even. We…

Q: 6) find all the and non residues (mods) incongruent quadratic residues

A: It is given that Incongurant quadratic residues.

Q: As a small illustration of Chinese remainder theorem that can all be solved mentally, say M = 30.…

A: The Chinese Remainder Theorem:                               Let m1,m2,.......mk be a collection of…

Q: If there are (2n+1) term in AP.then prove that the ratio of the sum of odd terms and the sum of even…

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Q: Prove by Strong Induction that, 2n-1+2n-2 = 3 x 2n-2 for every positive integer n.

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Q: Prove by contradiction that 8√2/ 7 is irrational. ( 8 root 2 by 7)

A: To prove  827  is irrational, We have to show that 2 is irrational, As we know that when multiple…

Q: then x* 1 is odd." Give a proof by contradicton to show "1fx s even integer,

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Q: Prove by using direct proof: Prove that the sum of any three consecutive integers is divisible…

A: Solution   Prove that the sum of any three consecutive integers is divisible by 3.

Q: () is even. 3. Prove that for all positive integers n, the number

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

A: 5 is a quadratic non- residue mod 17

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

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Q: Show that there are infinitely many primes to prove that there are infinitely many primes of the…

A: This question is related to Number Theory. We solve the result by contradiction.

Q: Suppose n ≥ 2 is an integer with the property that whenever a prime p divides n, p 2 also divides n…

A: let n=p1a1p2a2....pmam be the prime factorization of n  Here p1,p2....pm are all distinct primes and…

Q: Using the Law of Quadratic Reciprocity, prove that for an odd prime p ≠ 3 (3/p) = { 1 if p congruent…

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Q: Prove that the last two digits of 2" cannot be 02 and the last three digits cannot be 108. Find all…

A: All these problems can be solved using modular arithmetic. We presented the detailed solution in the…

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…

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Q: By Principle of Mathematical Induction prove 6n 6

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Q: 7. Use the Chinese remainder theorem to solve the following: 3x 2 (mod4) 5x =1 (mod 7) 8x 5 (mod9) I…

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Q: 2. Prove that V6 is irrational number.

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Q: "Every positive integer can be written as the sum of the square of 3 positive integers." prove…

A: Given: "Every positive integer can be written as the sum of the square of 3 positive integers." To…

Q: (a) Characterize the primes p for which 63 is a quadratic residue.

A: Quadratic residue: Let p be any odd prime member and gcd(a, p)=1. Then a is said to be a quadratic…

Q: using the Chinese Remainder Theorem how do I solve this systm of congruences?? 2x ≡ 1 (mod 5) 5x ≡ 4…

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Q: Find the smallest integer i for which the statement (1 + i2 )<2' is true. Then, prove the statement…

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Q: Prove that for any odd prime p different from 3, ) = 1 if and only if p = ±1 (mod 12). (hint: Use…

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Q: s) Express the infinite contimod fraction [2, 1,5 quadratic irrational- as a ity.

A: We need to express the infinite continued fraction 2, 1, 5¯ as a quadratic irrationality. The…

Q: How do I use the Chinese Remainder Thm to solve this system of congrunces below? x ≡ 1 (mod 3) x ≡ 4…

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Q: Solve by using the Chinese remainder theorem for the congruence X 2 (mod 7) X 4 (mod 5)

A: Given that This can be written as,

Q: Show that the product of two numbers of the form 4n + 1 is still of that form. Hence show that there…

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Q: 3. Use Euler's Theorem (and the Chinese Remainder Theorem) to how that n12 = 1 (mod 72) for all (n,…

A: The Euler's Theorem states that for a positive integer n and an integer a that is relatively prime…

Q: By using the Chinese remainder theorem, solve the following system of congruences x ≡ 4 (mod 11)…

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Q: By using mathematical induction prove that the given equation is true for all positive integers. 9n…

A: Steps to perform Mathematical Induction: 1) Consider an initial value (say n = 1) for which the…

Q: Prove that the number in between twin primes is always a multiple of 6.

A: A Twin prime are those numbers which are prime and having a difference of two ( 2 ) between the two…

Q: Let n = pq be the product of two odd primes. Recall that there are no primitive roots modulo n. Show…

A: We use the idea that if primitive root exists then x^2 = 1 mod (m) has only 2 solutions. The…

Q: Let p and q be odd primes. Prove that (p^2) q is not a perfect number. [Hint: Factor 1 + p + q + pq…

A: Let σp2q=σp2σq Then we have  σp2q=σp2σq=1+p+p21+q=p3-1p-1q2-1q-1≤p3p-1q2q-1

Q: Prove that no number of form 4k+3 (where K is a positive integer) can ever be the sum of two perfect…

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Q: Use the Division Algorithm to prove that for all positive integers, n,6|(n(n + 1)(2n + 1)).

A: We need to prove that, for all positive integers n, 6 is a factor of the expression given below.

Q: Prove that no integer congruent to 3 modulo 4 can be written as the sum of two squares.

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Q: Prove by Induction that for every natural number п, 3h-1 is a multiple of 2?

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Q: Use induction to prove that 16 divides 5n+2 + 13"+3 – 14 - for all n > 0.

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Q: Use the construction in the Chinese Remainder Theorem to solve the system of congruences x = 2 (mod…

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Q: Prove by direct method: for any n integer,if n is odd then 3n+2 is odd

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Q: Prove that the sum of two fourth powers of integers can never be three more than a multiple of 4. In…

A: Introduction: Divisibility is a property of an integer. For two integers a and b with b&gt;a, the…

Q: Prove that the difference of squares of any two odd numbers is múltiple of 8. (SOlve this task…

A: To prove- The difference of square of any two odd numbers is a multiple of 8.   Odd numbers are…

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.