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

Q: Let n be a positive integer. Prove 41+3n mod 9.

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Q: Let n E Z+, n > 2. Show that n is prime if and only if (n – 2)! = 1 (mod n).

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Q: 15. Prove that for each natural number n, 5+ ...+ (4n-3)(4n+1) 4n+1'

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Q: 6. Let p be an odd prime. Show that a2 = a (mod p*), k > 2 has exactly two solutions if x2 = a (mod…

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Q: Fermat's "Little" Theorem states that whenver n is prime and a is an integer, a"-l = 1 = 1 mod n a)…

A: Given a =463 and  n=953 We need to find x such that                   463952 ≡ x (mod 953).

Q: Find the smallest integer a > 1 such that a –1 = 2a (mod 11

A: To find:

Q: Use induction to prove that for every n E N, we have 5" = 1+ 4n (mod 8)

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Q: Prove that for all integers x, if x is not divisible by 7, then x^3 ≡ 1 mod 7 or x^3 ≡ −1 mod 7.

A: Since you have asked multiple questions, we will solve the Question Q2 for you. If youwant any…

Q: (a) Find three distinct integers satisfying the equation r = 1 mod 11. (b) Prove that if k is an…

A: a≡bmod m means a-b is divisible by m where a and b are integers. For example 13≡3 mod 5⇒13 - 3 is…

Q: Prove the statement by contradiction.

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Q: Let r be a primitive root of some n > 3. Prove that r = -1 (mod n).

A: Solution: Let n be any positive integer. Let Un= r : r is less than n and prime to n , where r=r+kn…

Q: Let n and a be positive integers and let d = gcd(a, n). Show that the equation ax mod n = 1 has a…

A:

Q: a) Let p be an odd prime. Show that a = -1 mod p has a solution if and only if p = 1 mod 8.

A: Consider the provided question,Hello. Since your question has multiple parts, we will solve first…

Q: (a) Let n be any integer. Prove that n2 is congruent to 0, 1 or 4 modulo 8.

A: Note: According to bartleby guidelines we have to answer only first question please upload the…

Q: Let n be a positive integer, and a an integer Prove that if the order of a modulo n is n-1 then n is…

A: Problem 2.  Solving it using the concept of phi function

Q: Let n be an integer greater than 1. If (n-2)! ≡ 1 mod n, then Prove n is a prime number.

A: Use the Wilson theorem and different case to solve this problem as in next step.

Q: 13. a. Prove that for every integer n 1, 10" = (-1)" (mod 11). |

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Q: 4. The least positive residue of 2^1000000 mod 17 is: * 1 None of these

A: Question 01: By Division algorithm,  10,00,000= 16*62,500 and since 17 is prime and 17 is not…

Q: Prove that an odd integer n is a square modulo every power of 2 if and only if n ≡8 1.

A: Given n is an odd integer. We have to prove that an odd integer n is a square modulo every power of…

Q: 3. Prove that if n is an odd integer then 7n+5 is even.

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Q: 7. Prove that if n is an odd positive integer or divisible by 4, then 13 +23 + 33 + .+ (n– 1)3 = 0…

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Q: Establish that if a is an odd integer, then 2" aʻ = 1 (mod 2" *2) for any n 2 1.

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Q: 2. Let p be a prime number such that p> 5. Prove that p2-1 =0 (mod 24).

A: To prove the below statement If p is a prime number and p > 5. p2−1≡0mod24

Q: 13. a. Prove that for every integer n > 1, 10" = (-1)" (mod 11).

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Q: 7. Let p be a prime number and n= 2P – 1. Prove that 2"-1 = 1 (mod n).

A: Let p be a prime number and n=2p-1. Prove that 2n-1≡1 (mod n)

Q: Prove that if gcd(a, n) = 1 and gcd(a – 1, n) = 1, then 1+ a + a² +a³ + ..+ a®(n)-1 = 0 (mod n).

A: we will prove that gcd (a, n) = 1, which  means that a and n are  prime. Let m= gcd (a, n) = gcd (a…

Q: Prove that for every integer n, n3 mod 6 = n mod 6.

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Q: Prove by induction that (2n+1)3 is an odd number for all positive integers n.

A: First principle mathematically induction : Suppose is a mathematical statement .Here n is a natural…

Q: Prove that (5n)!÷40^n×n! is natural number for every natural number n​

A: To Prove that (5n)!÷40^n×n! is natural number for every natural number n​

Q: Prove that if n is an integer and 3n + 2 is even, then n is even using a) a proof by contraposition

A: Given , n is an integer .To prove → if 3n+2 is even , then n is even using a proof by composition…

Q: If a has order n-1 mod n, then n is prime. O True O False

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Q: Prove that every prime greater than 3 can be written in the form 6n + 1 or 6n + 5.

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Q: Prove that if a1, a2, . . a, are n > 2 integers such that a; = 1 (mod 3) for every integer i (1 < i<…

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Q: 15. Prove that for each natural number n, s+ (4n-3)(4n+1) 4n+1 ...

A: Given The given expression is 11·5+15·9+·  ·  ·+14n-34n+1=n4n+1

Q: b) Prove that if n2 (mod 3) = 1 then n is not divisible by 3.

A: The required proof is as follows -

Q: For every integer n &gt; 1, prove that (n - 1)!mod n = n - 1 if and only if n is prime.

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Q: show that If n is a positive integer and a is any integer prime to n, then a ln) = 1(mod n), where…

A: It's called Euler's generalization to Fermats theorem. I have used integer m instead of n.

Q: Establish that if a is an odd integer, then 2" a“ = 1 (mod 2" *2) for any n 2 1.

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Q: Prove that for every odd prime 1000 < p < 2000 the number of –2, –1,2 which are squares mod p is not…

A: To prove that for every odd prime numbers strictly between 1000 and 2000 the numbers -2,-1, 2 which…

Q: Let p >2 be a prime. Prove that [(p-2) = (-1)/2 (mod p).

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Q: Prove the following:   a. If n is an integer and n + 3 is even, then n is odd by using a proof by…

A: Remark: Proof by contrapositive is based on the fact that an implication is equivalent to its…

Q: Suppose a and b are relatively prime positive integers. Is it true that (a+2b)4(ab) = 1 (mod ab) ?…

A: We try to solve the question using Euler's Theorem. The detailed solution typeset in LaTeX is…

Q: (b) Prove that if p is a prime greater than 2 then 2(p- 3)! = -1 mod p %3D

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Q: Prove that for all integers n, [n/7] = (n – 3)/7 if and only if n mod 7 = 3.

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Q: Let p be prime. Suppose ab≡0 mod p and a ≢ 0 mod p. Apply Euclid’s Lemma to prove b ≡0 mod p

A: Given that, Let p be prime. Suppose, ab≡0 mod p and a≢0 mod p. To prove: b≡0 mod p…

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: If p is a prime satisfying n < p < 2n, show that 2n = 0 (mod p)

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Q: (a) Consider the following statement: For all n € N, x(x − 1) = 0 mod n Which of "if", "only if",…

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Statements and Logic

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