problem with integers

Question

I am given the following problem in algebra:

 

Assume x and y are two odd numbers and x/y is an integer. Which of the following statements are true?

1. x + y is odd.

2. xy is odd.

3. x/y is odd.

4. x - y is odd.

 

My options are the following:

 

A. 1 and 2 only

B. 1 and 3 only

C. 2 and 3 only

D. 2, 3, and 4 only

E. 2 and 4 only

 

I would appreciate help in this matter. Thank you.

Answer 1

If x is odd and y is odd, (x+y) is even.  So, (1) is out.

 

xy is odd.  That makes (2) correct. 

  [example:  3*5=15; note that all even products have 2 as a factor, so either x or y has 2 as a factor and thus can't be odd as stated]

 

x/y is odd.  That makes (3) correct.  See explanation for (2) since division is multiplication by 1/number.

 

(x-y) is always even [difference between odd numbers is even].  That eliminates (4). 

 

So, (2) and (3) are correct.

 

Use lots of examples to solve this.

Answer 2

Pick any 2 odd numbers:

 

x = 9

y = 3

 

Now we test out each operator.

 

9 + 3 = 12     even

9 x 3 = 27      odd

9/3 = 3          odd  (integer)

9 - 3 = 6       even

 

 

Now which choices give you odd number answers?

 

 

Answer 3

Let us check these out one at a time:

 

1. x + y is odd.  FALSE.  The sum of 2 odd numbers is even.
2. xy is odd.  TRUE.  The product of 2 odd numbers is odd.
3. x/y is odd.  TRUE.  The ratio of 2 odd numbers is odd, if the ratio is an integer.
4. x - y is odd.  FALSE.  The difference of 2 odd numbers is even.

 

Only statements 2 and 3 are TRUE, so that makes (C) the correct answer.