The factorial operation ! applied to a positive integer n denotes the product of all integers greater than or equal to 1 and less than or equal to n. If k = 1! + 2! + 3! + . . . + p! , where p is a prime number greater than 10, what is the remainder when k is divided by 4?a)0b)1c)2d)3e)Cannot be determinedCorrect answer is option 'B'. Can you explain this answer?

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The factorial operation ! applied to a positive integer n denotes the product of all integers greater than or equal to 1 and less than or equal to n. If k = 1! + 2! + 3! + . . . + p! , where p is a prime number greater than 10, what is the remainder when k is divided by 4?

a)
0
b)
1
c)
2
d)
3
e)
Cannot be determined

Correct answer is option 'B'. Can you explain this answer?

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The factorial operation ! applied to a positive integer n denotes the ...

Given: k = 1! + 2! + 3! + . . . + p!

Applying the definition of factorial operation:

n! = 1 × 2 × 3 × 4 × 5 × 6 × 7. . . × (n-2) × (n-1) × n

where n is a positive integer

This implies that:

1! = 1
2! = 1 x 2
3! = 1 x 2 x 3
4! = 1 x 2 x 3 x 4
5! = 1 x 2 x 3 x 4 x 5
…
p! = 1 x 2 x 3 x 4 x 5 x 6 x 7 …x (p-2) x (p-1) x p

From 4! onwards, all terms are divisible by 4.

So, to find the remainder when k is divided by 4, we need to consider only terms till 3!

1! + 2! + 3! = 1+ 1 × 2 + 1 × 2 × 3 = 1 + 2 + 6 = 9

The remainder when 9 is divided by 4 is 1. So when k is divided by 4, the remainder is 1.

Answer: Option (B)

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Most Upvoted Answer

The factorial operation ! applied to a positive integer n denotes the ...

Understanding the Problem

To solve this problem, we need to understand the concept of factorials and how they relate to the remainder when divided by 4.

Factorial Definition

The factorial of a positive integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.

Divisibility Rule of 4

To find the remainder when a number is divided by 4, we only need to consider the last two digits of the number. If these two digits form a number divisible by 4, then the entire number is divisible by 4.

Solution

1. Since p is a prime number greater than 10, p! will end in 0 because it includes all integers from 1 to p, which includes 10 and all multiples of 10.
2. Similarly, (p-1)! will also end in 0 as it includes all numbers up to p-1, which also includes 10 and its multiples.
3. Continuing this pattern, we see that all factorials from 1! to p! will end in 0.
4. Therefore, the product k = 1! x 2! x 3!... x p! will end in 0.
5. When a number ends in 0, the remainder when divided by 4 is always 0.

Therefore, the remainder when k is divided by 4 is 0.

Conclusion

The correct answer is option A) 0.

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Community Answer

The factorial operation ! applied to a positive integer n denotes the ...

Given: k = 1! + 2! + 3! + . . . + p!

Applying the definition of factorial operation:

n! = 1 × 2 × 3 × 4 × 5 × 6 × 7. . . × (n-2) × (n-1) × n

where n is a positive integer

This implies that:

1! = 1
2! = 1 x 2
3! = 1 x 2 x 3
4! = 1 x 2 x 3 x 4
5! = 1 x 2 x 3 x 4 x 5
…
p! = 1 x 2 x 3 x 4 x 5 x 6 x 7 …x (p-2) x (p-1) x p

From 4! onwards, all terms are divisible by 4.

So, to find the remainder when k is divided by 4, we need to consider only terms till 3!

1! + 2! + 3! = 1+ 1 × 2 + 1 × 2 × 3 = 1 + 2 + 6 = 9

The remainder when 9 is divided by 4 is 1. So when k is divided by 4, the remainder is 1.

Answer: Option (B)

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