Prove that if a + x ≡ a +y (mod n), then x ≡y (mod n)

Q: Za) Let A = {n E Z | n = 0 (mod 2)} and B = {n € Z | n = 0 (mod 4)}. Prove that if n E (A – B), then…

A: Follow the steps below.

Q: Let p= 2k +1 be an odd prime number. Show that x* – 1 = (X – 1)(X – 2²) - · (X – k²) mod p.

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Q: 2. Let fk be the kth fibonacci number. Use strong mathematical induction to prove that for n2 3 fn >…

A: PROOF :

Q: Justifying your answer, find all integers x satisfying the two simultaneous congru- ences x = 1 (mod…

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Q: Use trial and error to find all replacements for x that are less than the modulus that make the…

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Q: Equivalence classes of mod 6 on Z have: a)  3 elements each b)  infinitely many elements each c)…

A: Concept Used: We know that ℤn be the set of all congruence class of ℤ for the relation congruence…

Q: Congruence conditions (mod 20) for primes p for which -5 is a square (mod p).

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Q: MAT 255 I. Let p and q be distinet prime mumbers and let a be an integer not divisible by p and not…

A: Note: Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: primitive root modulo

A: Given: Value of n=223 in  mod n. To find: Primitive root modulo 223.

Q: Prove that if n is a positive integer then n° z 2 mod 9. Show all your argument. Hint: use proof by…

A: Euclid-division lemma : Suppose a and b are two positive integer and when a is divided by b, then…

Q: Y OU (a) Find the least residue of 639 modulo 10. (b) Use Fermat's little theorem to find the least…

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Q: Using Fermat's Little Theorem find 52003 mod 455. Note that 455 is not a prime number and 455…

A: We are asked to use Fermat's Little theorem to find 52003 mod455. Fermat's Little Theorem states…

Q: DISCRETE MATHEMATICS  Prove that (a mod 2)(b mod 2) = (ab) mod 2. Use ANY METHOD of proof.

A: We need to prove that (a mod 2)(b mod 2) = (ab) mod 2

Q: 3. Congruence Let a, b, c > 0, m 2 2 be integers. Prove that a = b (mod m) implies that ac = bc (mod…

A: given that a≡bmod m where m≥2 a≡bmod mm/b-a (using definition) now m/b-a now using the result a/b…

Q: x and y are two positive integers such that y > 5x. Select the expressions that are not equal to…

A: We choose x and y such that we are able to discard options.

Q: *discrete math* Prove that for all integers a, b, if a and b are congruent to 1, mod 3, then their…

A: An integer n is congruent to 1, mod 3, if and only if there is an integer k such that n = 3k + 1.

Q: Number theory

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Q: Prove that n and m are natural numbers such that n > m, then n-m is also a natural number. Prove by…

A: Suppose that n and m are natural numbers such that n > m. Then we have n - m >0…

Q: Suppose n is a large odd number. You calculate 2(n-1)/2 = k (mod),  where k is some integer k is not…

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Q: all Natural numbers, n, that make 62 ≡≡ 46 mod n. Show all your work:

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Q: Let p = 2^k + 1 be a prime for some k ≥ 1. Let a be any integer such that gcd(a, p) = 1 and a is NOT…

A: Consider the given information:

Q: Let a and b be positive integers with (a, b) = 1. Show that if g is a primitive root (mod ab), then…

A: We have given (a,b)=1 where a, b both are positive integer. We have to show if g is primitive root…

Q: How do I prove this: Show that if x ≡ 1(mod n), then x^2 ≡ 1(mod n) and x^3 ≡ 1(mod n) and how to I…

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Q: Let odd prime number P =4 (mod L), be an ond If the integer g iso primitive root with respect to mod…

A: Let r be a primitive root of the prime p with p ≡ 1 (mod 4). We have to show that −r is also a…

Q: (a) Use cases based on congruence modulo 3 and properties of congruence to prove that for each…

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Q: Prove that ifa≡b(modn)andb≡c(modn), thena≡c(modn). Note that you must use thedivides definition of…

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Q: 28. Show that if p and q are distinct primes, then p-1+qP-=1 (mod pq).

A: To prove: If pand q are distinct primes then, pq-1+qp-1≡1(mod pq).

Q: 8. Let â be the inverse of a modulo k. Prove that the order of a modulo k is equal to the order of â…

A: If a is a primitive root modulo k , then for all n (coprime with k) there exists m , such that : am…

Q: mod n, then x ≡ 0 mod n or

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Q: Suppose g is a primitive root mod p (with p an odd prime) such that g^p-1 =/ 1 mod p+ Prove g must…

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Q: Show that if b0 = 0, then (g^xm)≡ 1 (mod p). Q2: Show that if b0 = 1, then (g^xm) ≡ p − 1 (mod p).

A: Given: x is a number in the set {0, 1, . . . , 2t} x written as: x = b0 · 20 + b1 · 21 + b2 · 22 + ·…

Q: Suppose p is a prime congruent to 3 modulo 4. (a) Prove that the congruence r = -1 (mod p) has no…

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Q: Fermat.

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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: Consider congruence mod 5. Let r = {0, 1, 2, 3, 4}. 1) Find r so that 3 =r mod (5). 2) Find r so…

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Q: Prove that there are infinitely many prime numbers that are square modulo 11 (trying to prove…

A: To prove: There are infinitely many prime numbers that are square modulo 11.

Q: Prove that the relation of congruence modulo n is transitive. That is, prove that for all integers…

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Q: Suppose n is a large odd number. You calculate 2(n-1)/2 = k (mod),  where k is some integer k is not…

A: Given that 2n-12≡kmodn Take the square on both sides, 2n-12≡kmodn2n-12·2≡k2modn2n-1≡k2modn (a) The…

Q: Use a direct proof: p is a prime # and x ∈ Z. If x2≡x (mod p) then x ≡ 0 (mod p) or x ≡ 1 (mod p).…

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Q: Use strong induction to prove that (Fibonacci number represented as Fn) Fn < 2^n for all n ∈ N

A: Let P(n) be the statement: Fn &lt; 2n . For n = 0, F0 = 0 and 20 = 1, therefore F0 &lt; 20. For n=…

Q: Let a, b, and c be integers and n a natural number. If a ≡ b mod n then a+c ≡ b+c mod n. How do I…

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Q: 4. Definition: Let M be a modulus greater than or equal to 2, and A an integer. If A" = 1(mod M) for…

A: Solution a: order of 4 Here 41≡4 (mod 13)42≡3 (mod 13)43≡12 (mod 13)44≡9 (mod 13)45≡10 (mod 13)46≡1…

Q: 3. Prove of disprove the following statements: (a) If a divides b and b divides a then a = b. (b) If…

A: Remark: let p and q be two integers, then we say that p divides q if there is a positive integer r…

Q: Using Fermat's Little Theorem, solve the congruence 2. x425 + 4 · x108 – 3 . x² + x – 4 = 0 mod 107.…

A: Consider the provided question, 2·x425+4·x108-3·x2+x-4≡0  mod 107                  ......1 Using…

Q: How do I do this:  1) Let p be a prime number and let a ∈ Z. Prove that p∣a if and only if a≡0(mod…

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Q: (4) (a) State Euler's Theorem or Fermat's Little Theorem. (b) Find the least non-negative residue of…

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Q: Proofs: prove that for all integers m and n, if n mod 5 = 3, then n2 mod 5 = 4

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Q: a. Among the numbers -14, -2, 2, 22, determine which pairs are congruent modulo 8. b. Using the…

A: Note: we know that a=b (mod n) if and only if a-b is divisible by n or in other words, we get same…

Statements and Logic

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