(Solved): Use the quotient-remainder theorem with divisor equal to 2 to prove the following statement. The s ...

Use the quotient-remainder theorem with divisor equal to 2 to prove the following statement. The square of any integer can be written in one of the two forms 4k or 4k+ 1 for some integer k. Proof: Suppose n is any integer. By the quotient-remainder theorem with divisor equal to 2, n = 2q or n = 2q + 1 for some ---Select--- ?q. Because n could have either one of these two forms, we must consider two separate cases. Case 1, n = 2q for some integer q: Consider the sentences in the following scrambled list. Then, by substitution, k is an integer with the property that n² = 4k. By substitution, n² = (29)² = 4g². By substitution, n² = (2g)² = 2g². By substitution, n² = 2(2g) = 4g. Nm & in Let k = Let k = q². Let k = q. Since n is an even integer and n = 2q, then q is an integer. Since q is an integer, then q2 is also an integer because it is a product of integers. Since q is an integer, then 29² is also an integer because it is a product of integers. By the assumption in Case 1, n = 2q for some integer q. To prove Case 1, select sentences from the list and put them in the correct order below. 1. --Select--- 2. --Select--- 3. --Select--- 2 4. --Select--- 5. -Select--- Case 2, n = 2q + 1 for some integer q: Consider the sentences in the following scrambled list. Now 4q² + 4g + 1 = 4(q² + g) + 1. Now 4q² + 4g + 1 = 4(q + 1) + 1. Now 49² + 4g + 1 = 4g(q² + q) + 1. Since q is an integer, then q² + q is also an integer because products and sums of integers are integers. Since q is an integer, then q(q² + q) is also an integer because products and sums of integers are integers. Since q is an integer, then q + 1 is also an integer because it is a sum of integers. Let k = q + 1. Let k = g(q² + q). Let k = q² + q. By substitution, n² = (2q + 1)² = 4q² + 4g + 1. Then, by substitution, k is an integer with the property that n² = 4k + 1. By the assumption in Case 2, n = 2q + 1 for some integer q. To prove Case 2, select sentences from the list and put them in the correct order below. 1. --Select--- 2. --Select--- 3. ---Select--- 4. --Select--- 5. --Select--- 6. --Select--- Conclusion: Since any integer n can be expressed in one of the ways shown in cases 1 and 2, we conclude that no matter what integer n is chosen, n can be written in one of the two forms specified in the given statement.