computer math 16 Consider He following Heave aiFXand are odd

computer math

16. Consider He following Heave aiFXand are_ odd nteas-es, -) Pru vide a direct Proof«F He hearth /7. Gue a?\".f by coutradiction ofHz Fallow,ng: ceiF n is an odd integer- H.ea nz is odd CC

Solution

16)

Direct Proof: Since x and y are both odd, we have that

x = 2a + 1 and y = 2b +1 for a, b integers.

Consider the sum x + y = (2a +1) + (2b + 1)

                                   = 2a + 2b + 2 = 2(a +b + 1) = 2k

where k is the integer a +b + 1. By definition we have that x + y is even

17)

Suppose n is odd and n² is even

so n=2k+1 and n²=2p

Therefore, consider (2k+1)²=2p

                            4k²+4k+1=2p.

                         2(2k2+2kp)=1

which shows even = odd, a contradiction by our assumption that n² is even.

Therefore n² is odd.