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If A is a positive integer, then which of the following statements is true?
1. A2 + A -1 is always even.
2. (A4+1)(A4+2) + 3A is even only when A is even.
3. (A-1)(A+2)(A+4) is never odd.
- a)1
- b)2
- c)3
- d)1 and 2
- e)2 and 3
Correct answer is option 'E'. Can you explain this answer?
Verified Answer
If A is a positive integer, then which of the following statements is ...
Step 1: Question statement and Inferences
We are given that A is a positive integer. A may be odd or even.
Step 2: Analyzing the given statements
1. A2 + A -1 is always even
Case (i) A is Odd
- A2 is also odd (Odd × Odd = Odd)
- A2 + A is even (Odd + Odd = Even)
- A2 + A -1 is odd ( Even – Odd = Odd)
We don’t even need to consider the case when A is even, because Statement 1 has already been proven FALSE.
2. (A4+1)(A4+2) + 3A is even only when A is even
Case (i): A is odd
- 3A is odd (Odd × Odd = Odd)
Now, an odd number multiplied with itself any number of times will give an odd product
- A4 is odd
- A4 + 1 is even (Odd + Odd = Even)
- A4 + 2 is odd (Odd + Even = Odd)
- (A4+1)(A4+2) + 3A = Even × Odd + Odd
- (A4+1)(A4+2) + 3A = Even + Odd
- (A4+1)(A4+2) + 3A = Odd
Thus, when A is odd, (A4+1)(A4+2) + 3A is odd
Case (ii): A is even
3A is even (Odd × Even = Even)
An even number multiplied with itself any number of times will give an even product
- A4 is even
- A4 + 1 is odd (Even + Odd = Odd)
- A4 + 2 is even (Even + Even = Even)
- (A4+1)(A4+2) + 3A = Odd × Even + Even
- (A4+1)(A4+2) + 3A = Even + Even
- (A4+1)(A4+2) + 3A = Even
Thus, when A is even, (A4+1)(A4+2) + 3A is even
Thus, Statement 2 is TRUE.
3. (A-1)(A+2)(A+4) is never odd
Case (i): A is odd
- A-1 is even (Odd -Odd = Even)
- A + 2 is odd (Odd + Even = Odd)
- A + 4 is odd (Odd + Even = Odd)
- (A-1)(A+2)(A+4) = Even × Odd × Odd
- (A-1)(A+2)(A+4) = Even
Thus, when A is odd, (A-1)(A+2)(A+4) is even
Case (ii): A is even
- A-1 is odd (Even -Odd = Odd)
- A + 2 is even (Even + Even = Even)
- A + 4 is even (Even + Even = Even)
- (A-1)(A+2)(A+4) = Odd × Even × Even
- (A-1)(A+2)(A+4) = Even
Thus, when A is even, (A-1)(A+2)(A+4) is even
Thus, Statement 3 is TRUE.
Answer: Option (E)
Most Upvoted Answer
If A is a positive integer, then which of the following statements is ...
To determine which of the given statements is true, let's evaluate each option one by one:
1. A^2 - A - 1 is always even:
- Let's test this statement with some positive integer values of A:
- A = 1: 1^2 - 1 - 1 = 1 - 1 - 1 = -1, which is odd.
- A = 2: 2^2 - 2 - 1 = 4 - 2 - 1 = 1, which is odd.
- A = 3: 3^2 - 3 - 1 = 9 - 3 - 1 = 5, which is odd.
- Based on these evaluations, we can conclude that the statement is not always true.
2. (A^4 + 1)(A^4 + 2) - 3A is even only when A is even:
- Let's evaluate this statement with some positive integer values of A:
- A = 1: (1^4 + 1)(1^4 + 2) - 3(1) = (1 + 1)(1 + 2) - 3 = 2 * 3 - 3 = 3, which is odd.
- A = 2: (2^4 + 1)(2^4 + 2) - 3(2) = (16 + 1)(16 + 2) - 6 = 17 * 18 - 6 = 306, which is even.
- A = 3: (3^4 + 1)(3^4 + 2) - 3(3) = (81 + 1)(81 + 2) - 9 = 82 * 83 - 9 = 6805, which is odd.
- Based on these evaluations, we can conclude that the statement is true.
3. (A-1)(A^2)(A^4) is never odd:
- Let's evaluate this statement with some positive integer values of A:
- A = 1: (1-1)(1^2)(1^4) = 0 * 1 * 1 = 0, which is even.
- A = 2: (2-1)(2^2)(2^4) = 1 * 4 * 16 = 64, which is even.
- A = 3: (3-1)(3^2)(3^4) = 2 * 9 * 81 = 1458, which is even.
- Based on these evaluations, we can conclude that the statement is true.
Therefore, the correct answer is option 'E' (2 and 3).
1. A^2 - A - 1 is always even:
- Let's test this statement with some positive integer values of A:
- A = 1: 1^2 - 1 - 1 = 1 - 1 - 1 = -1, which is odd.
- A = 2: 2^2 - 2 - 1 = 4 - 2 - 1 = 1, which is odd.
- A = 3: 3^2 - 3 - 1 = 9 - 3 - 1 = 5, which is odd.
- Based on these evaluations, we can conclude that the statement is not always true.
2. (A^4 + 1)(A^4 + 2) - 3A is even only when A is even:
- Let's evaluate this statement with some positive integer values of A:
- A = 1: (1^4 + 1)(1^4 + 2) - 3(1) = (1 + 1)(1 + 2) - 3 = 2 * 3 - 3 = 3, which is odd.
- A = 2: (2^4 + 1)(2^4 + 2) - 3(2) = (16 + 1)(16 + 2) - 6 = 17 * 18 - 6 = 306, which is even.
- A = 3: (3^4 + 1)(3^4 + 2) - 3(3) = (81 + 1)(81 + 2) - 9 = 82 * 83 - 9 = 6805, which is odd.
- Based on these evaluations, we can conclude that the statement is true.
3. (A-1)(A^2)(A^4) is never odd:
- Let's evaluate this statement with some positive integer values of A:
- A = 1: (1-1)(1^2)(1^4) = 0 * 1 * 1 = 0, which is even.
- A = 2: (2-1)(2^2)(2^4) = 1 * 4 * 16 = 64, which is even.
- A = 3: (3-1)(3^2)(3^4) = 2 * 9 * 81 = 1458, which is even.
- Based on these evaluations, we can conclude that the statement is true.
Therefore, the correct answer is option 'E' (2 and 3).
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