Integer x represents the product of all integers between 1 and 25, inclusive. The smallest prime factor of (x + 1) must be _____. a)between 1 and 10 ?b)between 11 and 20c)between 21 and 25d)greater than 25Correct answer is option 'D'. Can you explain this answer?

Question

Answer

Solution:
To find the smallest prime factor of (x+1), we need to find the value of x first.

Value of x:

x = 1 × 2 × 3 × … × 25

x = 25!

Now, we need to find the smallest prime factor of (x+1).

Smallest Prime Factor:

To find the smallest prime factor of (x+1), we need to add 1 to x and factorize it until we get the smallest prime factor.

x+1 = 25! + 1

Now, we need to factorize 25!+1 to find the smallest prime factor.

25!+1 = 2^k × m

where k is a non-negative integer and m is an odd integer.

We cannot factorize 25!+1 easily as it is a large number. However, we can use Wilson's theorem to simplify the expression.

Wilson's Theorem:

If p is a prime number, then (p-1)! ≡ -1 (mod p)

Using Wilson's theorem, we can say that:

24! ≡ -1 (mod 25)

Multiplying both sides by 25, we get:

25 × 24! ≡ -25 (mod 25)

25! ≡ -25 (mod 25)

25!+1 ≡ 0 (mod 25)

This means that 25 is a factor of 25!+1.

Now, we can say that:

25!+1 = 25 × k

where k is a positive integer.

Therefore, the smallest prime factor of (x+1) is greater than 25. Hence, the correct answer is option D.

Answer

According to me, the answer is D.

i think. if a number is a factor of x, it cannot be a factor of x + 1. so the smallest prime factor if greater than 25.

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