These questions are based on the following information:\n\nGiven $N = 1! + 2! + 3! + ........ + 99! + 100!$.\n\nFind the remainder when $N$ is divided by 168.

### Concept & Logic When finding the remainder of a factorial series divided by a composite number, break the divisor down into its prime factors. Find the smallest factorial that contains all these prime factors; every factorial after that will leave a remainder of 0. $$\text{Divisor} = 168$$ ### Step-by-Step Solution * **Given:** $N = 1! + 2! + 3! + ........ + 99! + 100!$. We need the remainder of $N / 168$. * **Calculation:** First, prime factorize the divisor, 168: $$168 = 8 \times 21 = 2^3 \times 3 \times 7$$ * We need to find the smallest factorial $n!$ that contains $2^3, 3,$ and $7$ as factors. * Look at $7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$. It obviously contains 7. It contains 3 (from the 3 and 6). It contains $2^4$ (from the 2, 4, and 6). * Since $7!$ is a perfect multiple of 168, every factorial from $7!$ onwards ($8!, 9!,$ etc.) will also be a multiple of 168. Thus, their remainders will all be 0. * We only need to evaluate the sum of the first six factorials: $$\text{Sum} = 1! + 2! + 3! + 4! + 5! + 6!$$ $$\text{Sum} = 1 + 2 + 6 + 24 + 120 + 720 = 873$$ * Now, simply divide this remaining sum by 168 to find the final remainder: $$168 \times 5 = 840$$ $$873 - 840 = 33$$ * The remainder is 33. ### Exam Strategy & Shortcut Whenever dividing a sum of factorials by a number, immediately prime factorize the divisor. Spot the largest prime factor (here, it is 7). That instantly tells you $7!$ is the cutoff point. You now know you only have to calculate up to $6!$. $6!$ is a standard memorized value (720). The sum is 873. A quick mental long division ($168 \times 5 = 840$) gives 33. ### Common Pitfall Students often get intimidated by the $100!$ and try to look for complex modular arithmetic cycles or Fermat's theorems. Factorial remainders are almost always a truncation problem. Don't overthink it; just find the cutoff where the divisor becomes a factor. ### Final Answer Therefore, the correct answer is **33**.