(a) Suppose n > 1 is a composite integer ab where a and b are unequal integers both greater than 1. Prove that (n – 1)! is congruent to 0 mod n. [Hint: Why are both factors less than n/2?] = p? (b) The preceding part of the problem proves the reverse implication unless n = where p is a prime. Prove that if p > 2 is prime then (p2 – 1)! is congruent to 0 mod p?, and find k E {0,1, 2, 3} such that (22 – 1)! is congruent to k mod 4.

Q: Prove that for all integers n ≥ 1, 10" = (-1)" (mod 11). Use part (a) to prove that a positive…

A: (a) Let n≥1 be an arbitrary integer. We know that 10≡10mod 11≡-1mod 11. Now consider 10n:…

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: Prove the following statement by contradiction. For any integer n, if n² + 4n + 5 is even, then n is…

A:

Q: (1) Give a proof by contradiction of the statement: If n? + 3n – 5 is even, then n is odd.

A:

Q: For any integer n ≥ 1, C(6n, 3n) C(6n, 3n-1). O True O False

A:

Q: Can you please show me step by step for this math that makes me undertandable. Q: Determine which…

A: (a) For the congruence 4x ≡ 5 (mod 6), we need to find a solution for x such that when 4x is…

Q: Identify the error in the following proof:“ If n amd m are two different odd integers then there…

A:

Q: Show that 2(p − 3)! + 1 ≡p 0, for all odd primes p.

A: Consider the given statement as follows. Show that 2p-3!+1≡0mod p, for all odd primes p. Consider an…

Q: Consider: for any integers a and b, if (a3 + b3) mod 3 does not equal 0, then (a + b) mod 3 does not…

A: The contrapositive of A implies B is not B implies not A. We use this and solve this problem.…

Q: Suppose that a, b are two integers such that 3a? – b3 = 1. Does it follow that a and b are…

A: Given that a, b are two integers such that 3a2-b3=1.

Q: Prove that n is odd if and only if 3n+9 is even. You must use definitions of even (and odd, if…

A: An even number is an integer that is divisible by 2, i.e., when divided by 2, the remainder is 0.An…

Q: If n is an integer, then (2n² + 3n)# 3(mod5).

A:

Q: 3. Prove the following statements: (a) If n is an even positive integer then n3 is an even positive…

A:

Q: For matrix A = (4 6 3 5 ). If f(x) = x2 - 5, calculate f(A)?

A: The given matrix is, A=4635. The function is f(x)=x2-5. For matrix multiplication: The number of…

Q: Prove that 3" < n! for all integers n 2 7.

A:

Q: Suppose n is an integer. Prove that if n is even, then n²+3n+5 is odd.

A:

Q: 1. Let a and b be natural numbers, where a, b > 2, such that gcd(a, b) = 1 and ab = (x + y)" for…

A:

Statements and Logic

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