If n = 1 + x, where x is the product of four consecutive positive integers, then which of the following is/are true ?
I. n is odd
II. n is prime
III. n is a perfect square
A. I only
B. I and II only
C. I and III only
D. None of these
Answer: Option C
Solution(By Examveda Team)
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 + (m2 + 3m) (m2 + 3m + 2)
⇒ n = 1 + y (y + 2), where m2 + 3m = y
⇒ n = 1 + y2 +2y = (1 + y)2 , which is a perfect square.
Hence, I and III are true.
Related Questions on Number System
Three numbers are in ratio 1 : 2 : 3 and HCF is 12. The numbers are:
A. 12, 24, 36
B. 11, 22, 33
C. 12, 24, 32
D. 5, 10, 15
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