Subject: General Aptitude · Chapter: Number System · Exam: · Marks: · Difficulty:
If n = 1 + x, where x is the product of four consecu- tive 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
Answer (as printed): C
Explanation
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.
Explanation as extracted from the printed page; notation may be imperfect.