Question

Every positive even integer is of the form 2q and every positive odd integer is of the form 2q+1, where q is some integer. True or false?
It's given true in the answer key but since it isn't specified that q is a 'positive' integer, I don't see how the answer can be right.

Answer 1

THE ANSWER WILL BE TRUE.

Answer 2

Answer:

If every positive odd integer is of the form 2q+1, then the even integer should be 2q but here it is given q.

Let a be any positive integer and b=2. Then, by Euclid’s division there exist integers q and r such that

A=2q+r, where 0≤r≤2

Now, 0≤r≤2→r=0 or r=1

.˙. a=2q or, a=2q+1

If a=2q, then a is an even integer.

We know that an integer can be either even or odd. Therefore, any odd integer is of the form 2q+1.