PROBLEM 4. If C is a (n, M, d) binary code where each codeword has odd weight, show that then d is even.
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Step 1: Assume that C is a (n,M,d) binary code where each codeword has odd weight. Show more…
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4) a) Using a proof by contraposition, prove that if n is an integer and 9n + 4 is odd, then n is odd. b) Using the Euclidean algorithm, determine the greatest common divisor of 9229 and 9977. You must show the details of your work.
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Prove that the following are equivalent for the integer $m$ : (a) $n$ is odd. (b) There exists $k \in \mathbf{Z}$ such that $n=2 k-1$. (c) $n^{2}+1$ is even.
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Problem 2. Prove ∀n, m ∈ Z, [ n³ + m³ odd ⟹ [n odd ∨ m odd] ] Hint: [P ⟹ [Q1 ∨ Q2] ] = [ [P ∧ (~ Q1)] ⟹ Q2] ]. Note (n³ + m³) - n³ = m³ and combine it with problem 1 - and think of what happens when you subtract (or add) 2 integers with different parity.
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