Question
When a counting number is added to 3 and the sum is divided by 2 the quotient will be an even number
Geometry
Choose the correct answer below
A. 3+2/2 = 5/2 which is not an even number
B. 8+4/2 = 6 which is an even number.
C. 10/2+3=9 which is not an even number.
D. there is no counterexample. The statement is correct
Geometry
Choose the correct answer below
A. 3+2/2 = 5/2 which is not an even number
B. 8+4/2 = 6 which is an even number.
C. 10/2+3=9 which is not an even number.
D. there is no counterexample. The statement is correct
Answers
GPT-4o mini
Let's analyze the statement and the options given carefully. The statement says that when a counting number \( n \) is added to 3 and the sum is divided by 2, the resulting quotient will be an even number.
We can express this mathematically:
\[
\frac{n + 3}{2}
\]
For the quotient to be an even number, \( n + 3 \) must be an even number, because dividing an even number by 2 results in an integer, and it must also yield an even integer.
1. If \( n \) is even, then \( n + 3 \) is odd (since odd + even = odd).
2. If \( n \) is odd, then \( n + 3 \) is even (since odd + odd = even).
From this, we conclude that \( \frac{n + 3}{2} \) will be an integer only when \( n + 3 \) is even, which happens when \( n \) is odd.
Now, let's evaluate the options provided:
A. \( 3 + 2/2 = 5/2 \) (not an even number)
- This calculation is incorrect because it should be \( (3 + 2)/2 = 5/2 \) which is not an even number.
B. \( 8 + 4/2 = 6 \) (an even number)
- This calculation is incorrect because it should be \( (8 + 3)/2 = 11/2 \), which is not an even number, and \( 8 \) is not the counting number used in the problem.
C. \( 10/2 + 3 = 9 \) (not an even number)
- This calculation doesn't match the exercise as it does not follow the structure provided.
D. There is no counterexample. The statement is correct.
- Based on our analysis, the statement is indeed correct when applied correctly with odd counting numbers.
**Conclusion**: The answer is **D**. However, it is important to clarify that the statement applies to odd counting numbers specifically.
We can express this mathematically:
\[
\frac{n + 3}{2}
\]
For the quotient to be an even number, \( n + 3 \) must be an even number, because dividing an even number by 2 results in an integer, and it must also yield an even integer.
1. If \( n \) is even, then \( n + 3 \) is odd (since odd + even = odd).
2. If \( n \) is odd, then \( n + 3 \) is even (since odd + odd = even).
From this, we conclude that \( \frac{n + 3}{2} \) will be an integer only when \( n + 3 \) is even, which happens when \( n \) is odd.
Now, let's evaluate the options provided:
A. \( 3 + 2/2 = 5/2 \) (not an even number)
- This calculation is incorrect because it should be \( (3 + 2)/2 = 5/2 \) which is not an even number.
B. \( 8 + 4/2 = 6 \) (an even number)
- This calculation is incorrect because it should be \( (8 + 3)/2 = 11/2 \), which is not an even number, and \( 8 \) is not the counting number used in the problem.
C. \( 10/2 + 3 = 9 \) (not an even number)
- This calculation doesn't match the exercise as it does not follow the structure provided.
D. There is no counterexample. The statement is correct.
- Based on our analysis, the statement is indeed correct when applied correctly with odd counting numbers.
**Conclusion**: The answer is **D**. However, it is important to clarify that the statement applies to odd counting numbers specifically.