Which of the following statements is FALSE? (a) 1526152716798897 % 5 = 2. (b) For every natural number а, if a2 % 4 = 0, then a % 5 = 1. (c) For every natural number a, if a % 6 = 5, then a = = 6q + 5 for some integer q. (d) For every natural number a, if a = b (mod 13), then a – b is divisible by 13.

Q: let 2^k+1 be a prime number. prove that then k=0 or k=2^n for some n =0

A: We have given that , 2k+1  be a prime number . We need to prove that , k = 0  or  k = 2n When k = 0…

Q: Prove by mathematical induction that for every positive integer n, 13 + 23 + 33 + .+ n³ n²(n+1)² is…

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Q: 13 + 33 + 53 +.…+ (2n – 1)3 = n²(2n² – 1) - ... Ensure that you explain the steps taken.

A: Prove by mathematical induction  13+23+33+…+2n-13=n22n2-1

Q: Prove that 01 + 02 is even.

A: Any odd integer, p can be represented as p=2k+1 . Any even integer, q can be represented as q=2k .…

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: For p an odd prime in Z , Prove: If p=(a)^2 + (b)^2  is possible in Z then   p= 1 (mod 4)

A: Compute general prime that satisfies given condition as follows.

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

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Q: For each of the following primes, find the smallest positive integer k such that k is a prim root…

A: A primitive root of a prime p is an integer g such that g(modp) has multiplicative order p-1.

Q: Give an inductive proof that the Fibonacci numbers Fn and Fn+1 are relatively prime for all n ≥ 0.…

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Q: fr2 = fnfn+ 1.r = 1

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Q: Determine for which positive integer values of n, 3n3 +2<n* and prove your claim by mathematical…

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Q: One number-theoretic result mentioned in the course was Wilson’s Theorem: If p is a prime then (p −…

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Q: For the first 5 odd primes, show that Wilsn's theorem holds to be true.

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Q: Hello, I need help for the problem in the picture. Thank you!

A: We are given that α ∈ Z is a primitive root modulo an odd prime p. So,    αϕ(p) ≡ 1 mod p =&gt; αp-1…

Q: Prove that 2" >n For all nuatural numbers

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Q: Use induction to show that n4 + 15 is a multiple of 16 for every positive odd integer n.

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Q: Prove the following using the mathematical induction Let c and d be integers. For every natural…

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Q: For which odd primes p is 5 a square modulo p?

A: As we know that  a is Quadratic residue mod p if and only if   a(p-1) /2 =1 mod p . If p and q are…

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: 111111 is prime. Prove that, in any radix b, The number 11111 (11...1), can be prime only if the…

A: As per the question we have to prove that the number 111...1 containing n number of 1's will be…

Q: 2. Find x such that 43177 = x (mod 39), 0≤x≤39. You must show all the detailed steps.

A: see my solutions

Q: Suppose n > 1 is a composite integer ab where a and b are unequal integers both greater than 1.…

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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: Given that P(n) is the equation 1+3+5+7+···+(2n−1) = n2, where n is an integer such that n ≥ 1, we…

A: Principal of mathematical induction

Q: 3. Prove that 2" > n For all nuatural numbers

A: Solution

Q: Let n be a positive integer. We write x; for the ith digit of n. (I.e., if n = 217, then x1 = 7, x2…

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Q: Is the following statement true or false? n+1 For every odd integer n, To answer this question, let…

A: Given statement true or fasle For every odd integer n {n /2} = n+1 /2

Q: For any natural number n, let Pn be the statement that n3 + n + 1 is even. i. Show that for every…

A: The given statement is Pn: n3+n+1 is even, where n is a natural number. i. Consider n∈ℕ. We have to…

Q: Prove for each integer n, that n2 is congruent either to 0 or to 1 modulo 4.

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Q: The preceding part of the problem proves the reverse implication unless n = p2 where p is a prime.…

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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: 12. Let the "Fibonacci-2" numbers g, be defined as follows: 91 = 2, 92 = 2, and gn+2 =( In+1)( In)…

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Q: In(n² = n(mod 10)), domain is the set of positive integers.

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Q: Prove the following using the mathematical induction Let c and d be integers. For every natural…

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Q: In the question that follows, the "units digit" of a number is the last digit in decimal…

A: 41 = 4 42 = 16 43 = 64 44 =256   Conjecture:- For all n&gt;=0, If n is odd then Unit digit of 4n is…

Q: In the question that follows, the "units digit" of a number is the last digit in decimal…

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Q: Prove the following statements by mathematical induction: - (a) Vn € N, - Σ (21 (2i − 1)² = n(2n +…

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Q: In the question that follows, the "Units Digit" or the "Unity Digit"of a number is the last digit in…

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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: A Wilson prime is a prime, p such that (p - 1)! = -1 (mod p2) For each of the following, determine…

A: Introduction: The only two elements in prime numbers are 1 and the number itself. When we consider…

Q: Prove the following:For every natural number n≥1, 1+4+7+⋯+(3n−2)=n(3n−1)2.

A: To prove the identity involving the sum to n terms of the given series

Q: For any integer n, either n2 mod 4 = 0, or n² mod 4 = 1.

A: For any integer n, we prove that either n2 mod4=0,  or n2 mod4=1

Q: (a) Suppose n > 1 is a composite integer ab where a and b are unequal integers both greater than 1.…

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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: Prove by induction that the sum of the first n odd numbers is n². In other words, prove E1(2i – 1) =…

A: To prove by induction that the sum of the first natural numbers is n^2.

Q: b) For which positive integer n is it true that 2"> n³? c) Prove your answer in (b) above using…

A: b. 2n&gt;n3 Now for n=1 21&gt;132&gt;1 For n=2 22&gt;234≯8 For n=3 23&gt;338≯27

Q: In the question that follows, the "units digit" of a number is the last digit in decimal…

A: We know 41 = 4, 42 = 16, 43 = 64, and 44 = 256

Q: 2. Suppose that instead of proving the claim "there is no integer n such that n= 2 mod 6 and n7 mod…

A: consider an integer n=26 such that n mod 6=26 mod 6               =2 mod 6 also, n mod 9=26 mod 9…

Q: 6. Show that if n (a2" - 1)/(a² - 1), where a is an integer, a > 1, and p is an odd prime not…

A: The given question is related with pseudoprimes. We have to show that if n = a2p - 1a2 - 1, where a…

Q: Let k be an integer such that k = -1 (mod 5). What is the remainder when 50k10 + 26k³ – 17 is…

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

Statements and Logic

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