Question

Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer​

Answer 1

QUESTION :-

Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer.

SOLUTION :-

Let a be any positive integer and b = 2. Then by division algorithm, a = 2q + r, for some integer q > 0, and r = 0 or r = 1.

Because 0 < r < 2. So, a = 2q or 2q + 1

If a is of the form 2q, then a is an even integer. Also, a positive integer can be either even or odd. Therefore any positive odd integer is of the form 2q + 1

Answer 2

Let ,

a = positive integer

b = 2

By division algorithm

a= 2q+r

q is greater than or equal to r

so,

a = 2q or 2q+1

If is the form of to give the knees knees and even integer also a positive integer can be either even or there for any positive integer is of the form of 2q+1