Suppose that a and b are integers such that a = 45 (mod 71) and b = 53 (mod 71). Find an integer c such that 0

Q: Prove that there are infinitely many primes congruent to 3 mod 8. previous part.

A: We have to show that there are infinitely many primes congruent to 3 mod 8. we will proof this by…

Q: TION 9 Prove by Induction that for every natural number n, 3"–1 is a multiple of 2? Attach File…

A: We prove by induction

Q: Let p be a prime. Show that -2 is a quadratic residue modulo p if and only if p = 1 mod 8 or p = 3…

A: Definition(Quadratic Residue Modulo): In number theory an integer qis called a quadratic residue…

Q: Find an integer x satisfying the equation: 4x21 = 2(mod 5) or show that no such integer exists.

A:

Q: 2n-1 b) Prove that if , r #1 (mod 2"), then n is even.

A: Given that x2n-1≢1mod2n . The objective is to show that n is even. The x2n-1≢1mod2n can be written…

Q: Show that 2 is a primitive root modulo 3" for all n > 1.

A: To prove that 2 is a primitive root modulo 3n for all n≥1.

Q: Show that if p is a prime number then every equation of the form ax ≡ b mod p with a 6≡ 0 mod p has…

A: Given p is a prime number and 6≡0 mod p. Therefore either p=2 or p=3. Let d=gcda,p. We know that if…

Q: What is the smallest integer that is a quadratic residue for all prime moduli p > 2?

A: We know if there is an integer 0<x<p, where p is a prime, such that x2≡qmod p, then q is said…

Q: Find all integer solutions satisfying all the equations given. 5x =1 mod 39 3x =1 mod 25 4x =1 mod…

A: Concept :.   *   ax =6 mod n                           g (d (a,n) = d (let)…

Q: Find x such that 15777 x (mod 11). 0sx<11. You must show all the detailed steps.

A: (3) We have to find the value of x such that 15777≡xmod11, 0≤x≤11. Since, 15777=14×11+377 Hence,…

Q: 24. Prove that for all integers m and n, if m mod 5 = 2 and n mod 5 = 1 then mn mod 5 = 2.

A:

Q: Let be a prime number. Stating any results that you use, deduce that xº-1 = (x – 1)(x – 2) · . · (x…

A:

Q: He have to Show that lon = (-1)" (mod 1") shall use mathematical moluction Let Pin) := lon a (47"…

A:

Q: (9) Find all prime numbers p for which both equations X²-2 0 mod p and have solutions. X²-3 0 mod p

A: Concept: The quadratic congruence: x2≡a(mod p) , where gcd(a,p)=1 Has a solution if (ap)=1 That is a…

Q: Find the least residue of  -5 -46 modulo 14.

A:

Q: Suppose n is an integer and consider the number n* – 6n³ – 18n² + 6n + 1. a) By expanding the…

A: Explanation: a. Given that,                    Expand the right side and show that both sides are…

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: 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: a) Show that 2340 ≡ 1 (mod 11) by Fermat’s little theorem and noting that 2340 = (210)34.b) Showthat…

A: According to the given information, Fermat’s little theorem says that:

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: Prove by any method(contradiction/contrapositive or cases)   Prove that (a mod 2)(b mod 2)= (ab)…

A:

Q: It is given that 3023 is prime, and 3023 = 2(1511) +1 where 1511 is also prime. (a) How many…

A:

Q: Suppose x satisfies the three simultaneous congruences X = 4 (mod 5) x = 7 (mod 8) X 10 (mod 11)…

A:

Q: Let 2k +1 be a prime number. Prove that then k = 0 or k = 2" for some n > 0. %3D

A: Solution:- Let 2k+1 be a prime number k=0 or k=2n for some n≥0   For, k=0 2k+1=20+1 2k+1=1+1…

Q: Determine all positive number replacements​ (less than the​ modulus) for the question mark that make…

A: Let the required number be x , where x&lt;1 .

Q: Find the smallest positive integer for which x mod 3=2 and x mod 4=3   What is the next smallest…

A:

Q: Prove by cases that for all integers n, n +5n is even. Case 1 (n is even): Case 2 (n is odd)…

A: The given term is n2+5n

Q: State whether True or False: If A and B are integers and A ≡ 3 (mod 3) and B ≡ 2 (mod 6) then A+B ≡…

A:

Q: 2. Find an inverse of 2 modulo 17 that is positive and less than 17.

A:

Q: Suppose x is a fixed real number with x 2 -1. For n2 1, prove using mathematical induction: (1+ x)"…

A: Here we will show that given statement is true by mathematical induction method.

Q: Verify by mathematical induction Ž (2i – 1)° = n°(2n² – 1) for alln2 1. - i=1

A:

Q: Suppose that 3x ≡ 2 (mod 5). Find the remainder when x is divided by 5; in other words, compute x…

A: ANSWER :

Q: Use Chinese theorem to find all positive integers that makes this congruence true:        x = 2 mod…

A:

Q: 1. Prove by the Principle of Mathematical Induction that 1 x 1! + 2 x 2! + 3 x 3! + ... +nxn! = (n +…

A: see below the explanation

Q: 25 Prove that mod 7 5 for all Gindegars and bin b mod 7-6

A:

Q: If n 2 1, then the Fermat number Fn 22" +1 is prime if and only if %3D 3(F.-1)/2 = -1 mod (Fn).

A:

Q: 6. Let f(x) be a polynomial with integral coefficients (like f(x) = 3x^^5-x^2+7) and suppose there…

A: Given:- fx=3x5-x2+7 There are 2 solutions  x mod 5 such that fx is congruent to 3 mod 5 and there…

Q: n>1 Problem 4. Prove by induction that, for every geq2 1 n +1 - - - 32 n2 2n

A:

Q: Use induction to prove for every positive integer, 4 x 10n = 4 or 7 (mod 33)

A: Suppose P(n) be any mathematical statement in terms of n. Then Principle of mathematical induction…

Q: Prove that by mathematically Induction (Marks = 05) 2n < (n + 1)!, for all integers n ≥ 2.

A: Answer

Q: Find all solutions to the congruence 10x≡5 mod 15 (b) Prove that 7 divides 32n+1+ 2n+2for any…

A: In  (a) we use the extended Euclidean algorithm to solve the problem. In (b) we use the mathematical…

Q: Prove that if n is an odd positive integer then 1 + 2 + 3 + · · · + (n − 1) ≡ 0 mod n . What happens…

A:

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: Suppose ab≡0 mod n and gcd(a, n) = 1. Apply Euclid’s Lemma to prove b≡0 mod n.

A: Given: ab≡0mod n and gcd(a,n)=1

Q: What is the smallest positive integer x in the equarion: 1=3x+2(mod 7)?

A: We have to solve the equation 1 = 3x + 2 (mod 7)⇒1 - 2 - 3x = 0 (mod 7)⇒-1 - 3x = 0 (mod 7)⇒3x = - 1…

Q: Number theory

A:

Q: 12x ≡ 18 (mod 15) has exactly 3 distinct solutions modulo 15 Select one: True False

A: 12x ≡ 18 (mod 15) has exactly 3 distinct solutions modulo 15 Select one: True False