9. [-/7 Points] DETAILS EPPDISCMATH5 4.2.027. Prove the following statement. The difference of any two odd integers is even. Proof: Let m and n be any odd integers. By definition of odd, there are integers r and s so that m can be expressed in terms of r and n can be expressed in terms of s as follows. Write m - n in terms of rand s and factor out a 2 to obtain m - n = Now is an integer because [---Select--- V of integers are integers. Therefore, m - n = 2 . (an integer), and so m - n is [---Select--- v by definition of [--Select--- v] . Need Help? Read It Watch it Submit Answer 10. [-/1 Points] DETAILS EPPDISCMATH5 4.2.034. Prove the following statement. Any product of four consecutive integers can be expressed as one less than a perfect square. (Two integers are consecutive if, and only if, one is one more than the other.) Proof: Consider any product of four consecutive integers. To simplify the computations, let n be the second smallest of the four. Then the product is (n - 1)n(n + 1) (n + 2) [ The aim is to try to express this product as the square of an integer minus 1. First express it as "something" minus 1 and then show that "something" is the square of an integer. ] Now (n - 1)n(n + 1)(n + 2) = n4 + (),3 - 12 - ( by multiplying out = 14 + () +3 - 12 - ) n + 1 ) - 1 = ( n2 + n - - 1 by algebra. Then (n - 1)n(n + 1)(n + 2) = m2 - 1, where m = Also, m is an integer because sums, products, and differences of integers are integers. Hence the given product of four consecutive integers can be expressed as one less than a perfect square. Need Help? Read It Submit