Find the unit digit of the multiplication of prime numbers between 100 to 1000

To find the unit digit of the multiplication of all prime numbers between 100 and 1000, we only need to consider the unit digits of these prime numbers. 

### Step 1: Identify the possible unit digits of prime numbers.
Prime numbers can only end in the digits 1, 3, 7, or 9 (except for the number 2 and 5, but those are not between 100 and 1000). Therefore, we only consider these unit digits.

### Step 2: Consider the behavior of multiplication of unit digits.
When multiplying numbers, the unit digit of the result only depends on the unit digits of the factors. Let's check the multiplication of these digits:

- \(1 \times 3 \times 7 \times 9 = 189\)

The unit digit of 189 is 9.

### Step 3: Multiply the numbers' unit digits
The key observation is that no matter how you multiply digits ending in 1, 3, 7, or 9, the pattern will repeat or return a digit from the set \{1, 3, 7, 9\}.

### Conclusion:
The unit digit of the multiplication of all prime numbers between 100 and 1000 will be **1**.

This is because as the multiplication involves many numbers, any sequence involving multiplication with the digit 9 will ultimately yield a unit digit of 1 when combined with the rest.

### Questions to Expand Your Understanding:

1. What is the role of modular arithmetic in finding unit digits?
2. How would the approach change if the range included numbers less than 100?
3. Why are prime numbers not ending in 0, 2, 4, 5, 6, or 8?
4. What would the process be for a smaller range, like between 10 and 100?
5. Can this technique be applied to find unit digits of non-prime sets?

### Tip:
Understanding modular arithmetic can simplify solving problems involving unit digits and remainders.

Discover how to find the unit digit of the multiplication of prime numbers between 100 and 1000 using modular arithmetic. This problem explores number theory and the properties of prime numbers, making it suitable for high school students in Grades 10-12.

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