What is the units digit of the product of all prime numbers between 1 and 100?

Options

A. 0
B. 1
C. 2
D. 3

Answer

To find the units digit of the product of all prime numbers between 1 and 100, we can follow these steps:

### Step 1: Identify the prime numbers between 1 and 100
The prime numbers between 1 and 100 are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

### Step 2: Focus on the units digit
To find the units digit of the product of these prime numbers, we can focus on the units digits of the numbers involved in the multiplication.

### Step 3: Identify key primes that affect the units digit
Notice that among the prime numbers, we have:
- 2 (which has a units digit of 2)
- 5 (which has a units digit of 5)

### Step 4: Multiply 2 and 5
When we multiply 2 and 5, we get:
\[ 2 \times 5 = 10 \]
The units digit of 10 is 0.

### Step 5: Consider the effect of multiplying by other primes
Now, if we multiply this result (10) by any other number, the units digit will remain 0. For example:
- \( 10 \times 3 = 30 \) (units digit is 0)
- \( 10 \times 7 = 70 \) (units digit is 0)
- Continuing this way, multiplying by any other prime will still yield a units digit of 0.

### Step 6: Conclusion
Since the product of all prime numbers includes both 2 and 5, the overall product will have a units digit of 0. Therefore, the units digit of the product of all prime numbers between 1 and 100 is:
\[ \text{Units digit} = 0 \]

### Final Answer
The units digit of the product of all prime numbers between 1 and 100 is **0**.

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