Unit Digit Of 0! + 1! + ... + 17! Sum

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Let's dive into finding the unit digit of the sum of factorials from 0! to 17!. This is a fun problem that involves understanding factorials and how they behave, especially when we're only interested in the last digit.

Understanding Factorials

First, let's quickly recap what a factorial is. The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example:

  • 0! = 1 (by definition)
  • 1! = 1
  • 2! = 2 × 1 = 2
  • 3! = 3 × 2 × 1 = 6
  • 4! = 4 × 3 × 2 × 1 = 24
  • 5! = 5 × 4 × 3 × 2 × 1 = 120

And so on. Now, let's look at how these factorials contribute to the unit digit of our sum.

Analyzing the Unit Digits

When we're trying to find the unit digit of a sum, we only need to focus on the unit digits of the numbers being added. Notice something interesting about factorials from 5! onwards:

  • 5! = 120
  • 6! = 6 × 5! = 720
  • 7! = 7 × 6! = 5040
  • 8! = 8 × 7! = 40320
  • 9! = 9 × 8! = 362880
  • 10! = 10 × 9! = 3628800

From 5! onwards, every factorial includes a factor of 5 and a factor of 2. This means that every factorial from 5! onwards will have a factor of 10, making the unit digit 0. This is super helpful because it simplifies our problem significantly! We only need to consider the factorials up to 4! when finding the unit digit of the sum.

Calculating the Sum of Unit Digits

So, we need to find the unit digit of the sum:

0! + 1! + 2! + 3! + 4!

Let's calculate these:

  • 0! = 1
  • 1! = 1
  • 2! = 2
  • 3! = 6
  • 4! = 24

Now, let's add the unit digits:

1 + 1 + 2 + 6 + 4 = 14

The unit digit of this sum is 4. Since all factorials from 5! to 17! have a unit digit of 0, they won't affect the unit digit of the total sum. Therefore, the unit digit of the sum 0! + 1! + 2! + 3! + ... + 17! is 4.

Conclusion

To summarize, we found that the unit digit of the sum 0! + 1! + 2! + 3! + ... + 17! is 4. We achieved this by recognizing that factorials from 5! onwards have a unit digit of 0, and then summing the unit digits of the factorials from 0! to 4!.

Extra Practice

To solidify your understanding, try similar problems. For example:

  1. What is the unit digit of the sum 1! + 2! + 3! + ... + 20! ?
  2. What is the unit digit of the sum 0! + 1! + 2! + 3! + ... + 25! ?

These problems can be solved using the same approach. Remember, once you hit 5!, the unit digits become 0, so you only need to focus on the initial factorials.

Why This Works

The reason this works is due to the properties of factorials and the decimal system. Once a factorial includes both 2 and 5 as factors, it becomes a multiple of 10. Multiples of 10 don't change the unit digit when added to other numbers. Understanding this principle can help you solve similar problems more efficiently.

Common Mistakes to Avoid

  1. Forgetting 0!: Remember that 0! is defined as 1. It's a common mistake to overlook this, but it's crucial for getting the correct answer.
  2. Calculating all factorials: Don't waste time calculating all factorials up to 17!. Once you realize that 5! and beyond have a unit digit of 0, you can stop there.
  3. Ignoring the unit digit concept: Focus only on the unit digits when summing. This simplifies the calculation and reduces the chance of errors.

Real-World Applications

While this specific problem might seem purely mathematical, the underlying principles are used in various fields:

  • Computer Science: Understanding factorials is crucial in algorithm design and analysis, especially in problems involving permutations and combinations.
  • Cryptography: Number theory, including properties of factorials, is used in cryptographic algorithms.
  • Statistics: Factorials are fundamental in probability calculations and statistical analysis.

Alternative Approaches

While the method described above is the most straightforward, here's another way to think about it:

Instead of calculating each factorial individually, you can keep track of the unit digit as you go:

  • 0! = 1 (unit digit is 1)
  • 1! = 1 (unit digit is 1)
  • 2! = 2 × 1! = 2 (unit digit is 2)
  • 3! = 3 × 2! = 6 (unit digit is 6)
  • 4! = 4 × 3! = 24 (unit digit is 4)
  • 5! = 5 × 4! = 120 (unit digit is 0)

And so on. This approach reinforces the idea that once you hit 5!, the unit digit will always be 0.

The Beauty of Math

Problems like these highlight the beauty of mathematics. They show how seemingly complex calculations can be simplified with the right understanding and approach. By breaking down the problem into smaller, manageable parts, we can arrive at the solution efficiently.

Keep practicing, and you'll become more comfortable with these types of problems. Remember to focus on the key concepts and look for patterns that can simplify your calculations. Happy problem-solving!

Final Thoughts

So, guys, remember the key takeaway: the unit digit of the sum 0! + 1! + 2! + 3! + ... + 17! is 4. This is because from 5! onwards, all factorials end in 0, and therefore don't affect the unit digit of the sum. Focus on the factorials from 0! to 4!, add their unit digits, and you'll have your answer! Keep exploring these mathematical concepts and you'll become a pro in no time!