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If m and v are integers
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05 Sep 2021, 15:02
05 Sep 2021, 15:02
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If \(m\) and \(v\) are integers and \(m^2 − v^2\) is an odd integer, which of the following must be an even integer?
I. \(mv\)
II. \(m − v\)
III. \(2m − v\)
(A) None
(B) I only
(C) II only
(D) III only
(E) I, II, and III
I. \(mv\)
II. \(m − v\)
III. \(2m − v\)
(A) None
(B) I only
(C) II only
(D) III only
(E) I, II, and III
Re: If m and v are integers
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05 Sep 2021, 19:07
05 Sep 2021, 19:07
1
Answer is (B) I only
if m^2 - v^2 is odd, which means either m or v is odd,
so mv is odd*even = even
m-v is diff between odd and even = odd
2m-v is inconclusive.
if m^2 - v^2 is odd, which means either m or v is odd,
so mv is odd*even = even
m-v is diff between odd and even = odd
2m-v is inconclusive.
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If m and v are integers
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10 Sep 2021, 08:25
10 Sep 2021, 08:25
1
Carcass wrote:
If \(m\) and \(v\) are integers and \(m^2 − v^2\) is an odd integer, which of the following must be an even integer?
I. \(mv\)
II. \(m − v\)
III. \(2m − v\)
(A) None
(B) I only
(C) II only
(D) III only
(E) I, II, and III
I. \(mv\)
II. \(m − v\)
III. \(2m − v\)
(A) None
(B) I only
(C) II only
(D) III only
(E) I, II, and III
Some important rules:
#1. ODD +/- ODD = EVEN
#2. ODD +/- EVEN = ODD
#3. EVEN +/- EVEN = EVEN
#4. (ODD)(ODD) = ODD
#5. (ODD)(EVEN) = EVEN
#6. (EVEN)(EVEN) = EVEN
Given: \(m^2 − v^2\) is odd
Factor to get: \((m+v)(m-v)\) is odd
So, it must be the case that \(m+v\) is odd, and \(m-v\) is odd.
If \(m+v\) is odd, then one of the values (m or v) must be odd and the other value must be even.
If one of the values (m or v) is odd and the other value is even, then for the product mv must be even, since one of the values is even.
So, statement I must be true.
This means we can eliminate answer choices A, C and D, since they state that statement I is not true.
Next, if one of the values (m or v) is odd and the other value is even, then we can be certain that m - v is ODD.
So, statement II is NOT true.
This means we can eliminate answer choice E, since it states that statement II is true.
By the process of elimination, the correct answer is B
Important point: Notice that I was able to solve the question without having to analyze statement III. So, once you've analyzed a statement, be sure to eliminate any answer choice that contradicts your analysis. In some cases, you'll find that you need not analyze all three statements.
