6. Use a direct proof to show that the sum of two odd integers is even.7. Use a direct proof to show that the sum of two even integers is even Show that if n is an integer andn3 + 5 is odd, then n is even usinga) a proof by contraposition.b) a proof by contradiction.8. Prove that if n is a positive integer, then n is odd if and only if 5n + 6 is odd.9. Find a counterexample to the statement that every positive integer can be written as the sum of thesquares of three integers.
Question
- Use a direct proof to show that the sum of two odd integers is even.7. Use a direct proof to show that the sum of two even integers is even Show that if n is an integer andn3 + 5 is odd, then n is even usinga) a proof by contraposition.b) a proof by contradiction.8. Prove that if n is a positive integer, then n is odd if and only if 5n + 6 is odd.9. Find a counterexample to the statement that every positive integer can be written as the sum of thesquares of three integers.
Solution
Sure, let's go through these one by one:
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Direct proof for the sum of two odd integers being even: An odd number can be expressed as 2k+1 where k is any integer. So, let's take two odd numbers, say a and b. a = 2m+1 and b = 2n+1 where m and n are integers. The sum of a and b is a+b = (2m+1) + (2n+1) = 2m + 2n + 2 = 2(m+n+1). As (m+n+1) is an integer, we can say that a+b is a multiple of 2, hence even.
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Direct proof for the sum of two even integers being even: An even number can be expressed as 2k where k is any integer. So, let's take two even numbers, say a and b. a = 2m and b = 2n where m and n are integers. The sum of a and b is a+b = 2m + 2n = 2(m+n). As (m+n) is an integer, we can say that a+b is a multiple of 2, hence even.
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Proof that if n is a positive integer, then n is odd if and only if 5n + 6 is odd: If n is odd, then n can be expressed as 2k+1 where k is an integer. So, 5n+6 = 5(2k+1)+6 = 10k+5+6 = 10k+11 = 2(5k+5)+1, which is an odd number. Conversely, if 5n+6 is odd, then it can be expressed as 2m+1 where m is an integer. So, 5n = 2m+1-6 = 2m-5. As 5n is a multiple of 5, 2m-5 must also be a multiple of 5. Therefore, n = (2m-5)/5 = 2(m/5) - 1, which is an odd number.
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Counterexample to the statement that every positive integer can be written as the sum of the squares of three integers: The number 7 is a counterexample. There are no three integers a, b, and c such that 7 = a^2 + b^2 + c^2.
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