Odd and even interger

Question

If P is an odd integer, and if Q is an even integer, which of the following expressions cannot represent an even integer? 

(1) 3P-Q

(2) 3PxQ

(3) 2QxP

(4) 3Q-2P

(5) 3P-2Q

 

i know the answer is (1), I don't understand why (5) can represent an even integer? If 2  times even interger is even interger, odd interger subtract even interger is not odd interger?

Answer 1

if P is an odd number we can write this way:

P=2*k+1

if Q is an even integer we can find an integer n, so we can write Q= 2*n

So let's try each expression:

1) 3*P-Q=6*k+3-2*n=2*(3*k+1-n)+1  which is an odd integer

2) 3*P*Q=3*(2*k+1)(2*n)=3(6*k*n+2*n)=2*(9*k*n+3) which is an even number

3) 2*Q*P=2*(2*n)(2*k+1) this integer is an even number it's obvious because it's equal the product of the number 2 with an integer.

4) 3*Q-2*P= 6*n -4*k-2=2*(3*n-2*k-1) which is an even number 

5) 3*P-2*Q=6*k+3-4*n=2*(3*k-2*n+1)+1, which is an odd integer.

 

We want to summarize there are two odds integer, which are:

3P-Q and 3P-2Q

 

Answer 2

Even + even = even

 

Odd + odd = even

 

Odd + even = odd.

 

 

Even times even = even

 

Even times odd = even

 

odd times odd = =odd.

 

 

If P is odd and Q is even, then

 

(1)  3P - Q is odd - even = odd

 

(2)  3P times Q is odd times even = even

 

(3)  2Q times P is even times odd = even

 

(4)  3Q - 2P is even - even = even

 

(5)  3P - 2Q is odd - even = odd

 

 

Expression (5) cannot represent an even integer, either.  If (5) were 3P - (1/2)Q, it would be even whenever Q was the double of an odd integer.