SOLUTION: If p and r are positive integers and 2p + r -1 = 2r + p + 1, which of the following must be true? I. p and r are consecutive integers. II. p is even. III. r is odd.

Question 307510: If p and r are positive integers and 2p + r -1 = 2r + p + 1, which of the following must be true?
I. p and r are consecutive integers.
II. p is even.
III. r is odd.

(A) None
(B) I only
(C) II only
(D) III only
(E) I, II, and III

Answer by rapaljer(4671) (Show Source): You can put this solution on YOUR website!
2p+r-1=2r+p+1 can be simplified by subtracting p and r from each side.
2p+r-1-p-r=2r+p+1-p-r
p-1=r+1

Now, add +1 to each side:
p=r+2

This means that p and r are NOT consecutive integers. It does indicated that if r is even, then p is also even. If r is odd, then p is also odd. The numbers are either BOTH even or BOTH odd. However, you can't say that p is necessarily even or odd, and you can't say that r is necessarily even or odd.

The correct answer is A) NONE.

Dr. Robert J. Rapalje, Retired
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