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If m, n, and p are positive integers such that m + n is even and the value of (m + n)2 + n + p is odd, which of the following must be true?
- a)m is odd
- b)n is even
- c)p is odd
- d)If n is even, p is odd
- e)If p is odd, n is odd
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
If m, n, and p are positive integers such that m + n is even and the v...
You are given that m + n is even and the value of (m + n)2 is also even. However, because (m + n)2 + n + p is odd, the sum n + p must be odd. A sum of two positive integers is odd only when one is even and one is odd. Therefore, it must be true that if n is even, p is odd.
Community Answer
If m, n, and p are positive integers such that m + n is even and the v...
To determine which of the given options must be true, let's analyze the given information step by step.
Given:
- m, n, and p are positive integers
- m + n is even
- (m + n)^2 + n + p is odd
1. m + n is even:
- The sum of two positive integers is even if and only if both integers have the same parity (either both are even or both are odd).
- Therefore, we can conclude that either both m and n are even or both are odd.
2. (m + n)^2 + n + p is odd:
- The square of an even number is always even, and the sum of even and odd numbers is always odd.
- Since (m + n)^2 is even and n is added to it, the sum becomes odd.
- Adding p to an odd number will also result in an odd number.
- Therefore, we can conclude that p must be odd.
Now, let's evaluate each option based on the information we have:
a) m is odd:
- We know that either both m and n are even or both are odd.
- Therefore, we cannot determine the parity of m solely based on the given information.
- Option a) cannot be determined as true.
b) n is even:
- We know that either both m and n are even or both are odd.
- The given information does not provide any specific information about the parity of n.
- Option b) cannot be determined as true.
c) p is odd:
- We have already determined that p must be odd based on the given information.
- Option c) is true.
d) If n is even, p is odd:
- We have determined that p must be odd regardless of the parity of n.
- Option d) is true.
e) If p is odd, n is odd:
- We know that either both m and n are even or both are odd.
- However, the given information does not provide any specific information about the parity of n.
- Option e) cannot be determined as true.
Based on the analysis, options c) and d) are the only ones that can be determined as true.
Given:
- m, n, and p are positive integers
- m + n is even
- (m + n)^2 + n + p is odd
1. m + n is even:
- The sum of two positive integers is even if and only if both integers have the same parity (either both are even or both are odd).
- Therefore, we can conclude that either both m and n are even or both are odd.
2. (m + n)^2 + n + p is odd:
- The square of an even number is always even, and the sum of even and odd numbers is always odd.
- Since (m + n)^2 is even and n is added to it, the sum becomes odd.
- Adding p to an odd number will also result in an odd number.
- Therefore, we can conclude that p must be odd.
Now, let's evaluate each option based on the information we have:
a) m is odd:
- We know that either both m and n are even or both are odd.
- Therefore, we cannot determine the parity of m solely based on the given information.
- Option a) cannot be determined as true.
b) n is even:
- We know that either both m and n are even or both are odd.
- The given information does not provide any specific information about the parity of n.
- Option b) cannot be determined as true.
c) p is odd:
- We have already determined that p must be odd based on the given information.
- Option c) is true.
d) If n is even, p is odd:
- We have determined that p must be odd regardless of the parity of n.
- Option d) is true.
e) If p is odd, n is odd:
- We know that either both m and n are even or both are odd.
- However, the given information does not provide any specific information about the parity of n.
- Option e) cannot be determined as true.
Based on the analysis, options c) and d) are the only ones that can be determined as true.
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If m, n, and p are positive integers such that m + n is even and the value of (m + n)2 + n + p is odd, which of the following must be true?a)m is oddb)n is evenc)p is oddd)If n is even, p is odde)If p is odd, n is oddCorrect answer is option 'D'. Can you explain this answer?
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