The problem presented asks for the last digit of the multiplication of all prime numbers between 800 and 1000.

Step-by-Step Solution:

1. Identify Prime Numbers between 800 and 1000: The prime numbers within this range are:

   • 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997.

2. Multiplication and Modulo Operation: The key to solving the problem is to find the last digit of the product of these primes. However, instead of calculating the entire product (which is computationally intense), you can use modular arithmetic to find the last digit.

3. Simplification:

   • Notice that one of the primes in the list is 5 (if it were there), multiplying by 5 would result in a number ending in 0 when multiplied by any even number. But since 5 is not a prime in this range, we move on to multiplying the primes modulo 10.
   • The multiplication of all these prime numbers modulo 10 will result in: $(9 \times 1 \times 9 \times 3 \times \ldots) \mod 10$
   • Continuing this calculation, you'll find that any number multiplied by 0 will yield a last digit of 0.

4. Conclusion: Since the product involves a 0, the last digit of the entire product of these primes is 0.

Would you like more detailed explanations or have any questions? Here are some related questions you might find interesting:

1. What is modular arithmetic and how does it apply in finding the last digit of products?
2. How do you determine whether a number is prime within a given range?
3. How does the presence of the number 5 affect the multiplication of numbers in terms of their last digit?
4. Can the concept of modular arithmetic be used to solve other types of problems involving large products?
5. How do you estimate the number of prime numbers within a certain range?

Tip: Understanding modular arithmetic is very useful in solving problems involving large numbers, especially when you are interested in only a specific part of the result, like the last digit or remainder.