Question

If n=1+x, where x is the product of 4 consecutive positive integers, then which of the following is/are true?
1. n is odd;
2. n is prime
3. n is a perfect square

• A. 1 and 3 only
• B. 1 and 2 only
• C. 1 only
• D. None of these

Answer 1

The Correct Option is C

To determine which statements about n are true, let's analyze the given condition:

Step 1: Define the problem. We have n = 1 + x, where x is the product of 4 consecutive positive integers, i.e., x = k(k+1)(k+2)(k+3) for some positive integer k.

Step 2: Check if n is odd. Since x = k(k+1)(k+2)(k+3) includes 4 consecutive integers, at least one of these integers is even, making x even. Therefore, x is an even number, making 1 + x an odd number because adding 1 to an even number results in an odd number. So, statement 1 is true.

Step 3: Check if n is prime. For n = 1 + x to be prime, it should have no positive divisors other than 1 and itself. However, since x is a product of 4 consecutive integers, it is always greater than 1. Thus, n = 1 + x will never represent a prime number because it results in an integer that can be divided by other numbers. Hence, statement 2 is false.

Step 4: Check if n is a perfect square. A perfect square is an integer that is the square of an integer. Since x is the product of 4 consecutive integers, x increases rapidly, and consequently, 1 + x typically doesn't form perfect squares due to inconsistent patterns. Therefore, statement 3 is false.

Ultimately, only statement 1 is true, so the correct answer is 1 only.