Question

Set s consists of the integers of the form r^2 +s , where r and s are integers such that . How many integers in s are odd?

Answer 1

Set s= {$r^2 +s$: is an Integer, r and s are integers.}

Following are the Possibilities

1. If r is even integer and s is even integer,Sum of Square of an even integer and even integer will be even.

2. If r is odd integer and s is odd integer then sum of Square of odd integer and and odd integer will be an even integer.

3. If r is even integer and s is an odd integer then sum of square of an even integer and an odd integer will be an Odd integer.

4. If r is an odd integer and s is an even integer then sum of square of an odd integer and an even integer will be an odd integer.

There are two cases in which Set s will be Odd:

a. If r is even integer and s is an odd integer then sum of square of an even integer and an odd integer will be an Odd integer.

b. If r is an odd integer and s is an even integer then sum of square of an odd integer and an even integer will be an odd integer.