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Which of the following is a valid proof for the statement: If n is an even integ

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Question: Which of the following is a valid proof for the statement: If n is an even integer, then n^2 is even?

Options:

  1. n = 2k for some integer k
  2. n = 2k + 1
  3. n = k^2
  4. n = 2k - 1

Correct Answer: n = 2k for some integer k

Solution: 

If n = 2k, then n^2 = (2k)^2 = 4k^2, which is even.

Which of the following is a valid proof for the statement: If n is an even integ

Practice Questions

Q1
Which of the following is a valid proof for the statement: If n is an even integer, then n^2 is even?
  1. n = 2k for some integer k
  2. n = 2k + 1
  3. n = k^2
  4. n = 2k - 1

Questions & Step-by-Step Solutions

Which of the following is a valid proof for the statement: If n is an even integer, then n^2 is even?
Correct Answer: If n = 2k, then n^2 = (2k)^2 = 4k^2, which is even.
  • Step 1: Understand what an even integer is. An even integer can be written as n = 2k, where k is any integer.
  • Step 2: Substitute n in the expression n^2. If n = 2k, then n^2 = (2k)^2.
  • Step 3: Calculate (2k)^2. This equals 4k^2.
  • Step 4: Recognize that 4k^2 is even. Since 4 is an even number and multiplying it by any integer (k^2) will also result in an even number.
  • Step 5: Conclude that if n is an even integer, then n^2 is also even.
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