MCQ

If n = 1 + x, where x is the product of four consecutive positive integers, then which of the following is true?

I. n is an even number

II. n is a prime number

III. n is a perfect square

IV. n is odd number

II. n is a prime number

III. n is a perfect square

IV. n is odd number

#### Options

I, III

I, II

I, IV

III, IV

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#### Solution

**III, IV**

**Explanation:**

We know that product of any two consecutive positive integers is always even number.

So, X = product of any four consecutive positive integers is always even number.

Now, n = (1 + x), is alway a odd number

Let x = 1.2.3.4 = 24

n = 1 + 24 = 25, which is a perfect square

again, for, x = 2.3.4.5 = 120

n = 1 + 120 = 121, a perfect square.

And this will follow for all positive integer value.

Hence, n is a perfect square odd number.

Concept: Number System (Entrance Exam)

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