Question

Suppose a,b are integers.If ab is even and a+b is odd, then only one of a and b is even.

a) Symbolise it

b) Give its contraposition in symbolic form

c) Translate your answer in b) back into English.

Answer 1

A.

ab=a(a+b−a)=a(a+b)-a² and since ab and a+b are even then a² is even so a is even. a and b play a symmetric role so bb is also even.

B.We can prove the statement by proving its contrapositive:

"If aa and bb are not both even, then abab and (a+b)(a+b) are not both even.

I.e., if either a or b is odd, then either ab or a+b is odd.

.Suppose a is odd. Then a=2k for some integer k. Then ab=(2k+1) b=2kb+b which is odd if b is odd, even otherwise. And a+b=2k+1+b which is odd if b is even, and even otherwise. So regardless of whether b is even or odd, (it must be one or the other), one of (a+b) or ab is necessarily odd.

The argument is identical if we suppose b is odd.

C. If ab is even, then at least one of a and b is even. Suppose a is even. Then if b is even (a+b) is even, while if b is odd then (a+b) is odd. Hence a and b are both even.