We recognize the expression for  as a perfect square trinomial which can be rewritten as:

 is odd if and only if the square root  is odd. This happens if and only if one of  and  is even and one is odd - this is the correct response.

If  and  are both odd, then , the product of odd numbers, is odd, and , the sum of three odd numbers, is odd.

If  and  are both even, then , the product of even numbers, is even, and , the sum of three even numbers, is even.

If one of  and  is even and one is odd, then , which has an even factor, is even, and , the sum of two even numbers and an odd number, is odd.

Therefore,  is even if and only if  and  are both even.

For the product of two positive integers to be odd, both of the integers must themelves be odd. Therefore, if is odd, it follows that both and are odd as well. We examine each of the quantities keeping this in mind.

(a) Both and are odd, so  and  are even. Their product, which is , must be even.

(b) Since  is odd, both  and  are even. Their product,   , is even.

(c) Since both and are odd, their squares and are both odd; it follows that  and  are both even, so their product is even.

The correct response is that all of the three quantites must be even.

is even, so it follows that one of and is even; the other can be even or odd. Let us assume that will always be even; the argument will be similar if is assumed to always be even.

(a)  is even, so is odd. If is even, then  is odd, and , the product of odd factors, is odd. If is odd, then  is even, and , having an even factor, is even. Therefore, can be even or odd.

(b) is even, so and are both odd. is the product of odd factors and must be odd.

(c) is even, so is even as well, and is odd. If is even, then  is even, and   is odd; , the product of odd factors, is odd. If is odd, then  is odd, and  is even; , having an even factor, is even. Therefore,  can be even or odd.

The correct response is that only (b) need be odd.

is an odd quantity if and only if one of and is even and the other is odd. We can assume without loss of generality that is the even quantity and is the odd quantity, since a similar argument holds if is even.

(a) is odd, so is even. , which has an even factor, is even.

(b)  is even and  is odd, so  is even, and and are both odd. , the product of odd factors, is odd.

(c)  is odd, so  is odd, and is even. , which has an even factor, is even.

The correct response is (a) and (c) only.

Which of the following is not an integer?

The definition of an integer is, "All positive or negative whole numbers, including zero."

Therefore, our answer must pi, because pi is not a whole number. All other options fit the defintion of integers.

An integer is an positive or negative whole number, including zero. Eliminate all options which are not whole numbers and you are left with 0!

We know that  is a positive integer with an even factors of prime numbers, for example  could be  or  could be , in other words,  must be a perfect square. The only possible answer is , the perfect square of .