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03 Apr 2020, 22:40
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D
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Difficulty:
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38% (00:55) correct
63%
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If n = 1 + x, where x is the product of four consecutive positive integers, then which of the following is/are true?
A. n is odd
B. n is prime
C. n is a perfect square
A. A only
B. B only
C. A and B only
D. A and C only
E. A, B and C
A. n is odd
B. n is prime
C. n is a perfect square
A. A only
B. B only
C. A and B only
D. A and C only
E. A, B and C
Archived Topic
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03 Apr 2020, 22:57
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n = 1 + x, where x is the product of four consecutive positive integers,
Let n = 1 + x = 1 + m (m + 1) (m + 2) (m + 3), where m is a positive integer.
Then, clearly two of m, (m + 1), (m + 2), (m + 3) are even and so their product is even.
Thus, x is even and hence n = 1 + x is odd
Also, n = 1 + m (m + 3) (m + 1) (m + 2) = 1 + (m^2 + 3m) (m^2 + 3m + 2)
⇒ n = 1 + y (y + 2), where m2 + 3m = y
⇒ n = 1 + y2 +2y = (1 + y)2 , which is a perfect square.
Also, a perfect square cannot be a prime number.
Hence, I and III are true.
correct answer is D
Let n = 1 + x = 1 + m (m + 1) (m + 2) (m + 3), where m is a positive integer.
Then, clearly two of m, (m + 1), (m + 2), (m + 3) are even and so their product is even.
Thus, x is even and hence n = 1 + x is odd
Also, n = 1 + m (m + 3) (m + 1) (m + 2) = 1 + (m^2 + 3m) (m^2 + 3m + 2)
⇒ n = 1 + y (y + 2), where m2 + 3m = y
⇒ n = 1 + y2 +2y = (1 + y)2 , which is a perfect square.
Also, a perfect square cannot be a prime number.
Hence, I and III are true.
correct answer is D
04 Apr 2020, 00:26
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Dillesh4096
If n = 1 + x, where x is the product of four consecutive positive integers, then which of the following is/are true?
A. n is odd
B. n is prime
C. n is a perfect square
A. A only
B. B only
C. A and B only
D. A and C only
E. A, B and C
A. n is odd
B. n is prime
C. n is a perfect square
A. A only
B. B only
C. A and B only
D. A and C only
E. A, B and C
Discussed here: if-n-1-x-where-x-is-the-product-of-four-consecutive-positive-inte-287581.html?fl=similar
THE TOPIC IS LOCKED AND ARCHIVED.
Archived Topic
Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
Where to now? Join ongoing discussions on thousands of quality questions in our Problem Solving (PS) Forum
Still interested in this question? Check out the "Best Topics" block above for a better discussion on this exact question, as well as several more related questions.
Thank you for understanding, and happy exploring!
