The set of all the prime num hers which are greater then 2 but less than 222, Find the unit digit in the product of the set ?

Options

A. 4
B. 5
C. 0
D. None of these

Answer

To find the unit digit in the product of all prime numbers greater than 2 but less than 222, follow these steps:

### Step 1: Identify the prime numbers in the given range
The prime numbers greater than 2 and less than 222 are:
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223.

### Step 2: Analyze the product of these prime numbers
When calculating the product of these prime numbers, we can observe that:
- The prime number 5 is included in the set.
- The prime number 2 is not included since we are only considering primes greater than 2.

### Step 3: Determine the unit digit of the product
The presence of the prime number 5 in the product means that the product will have a unit digit of 0 if there is at least one even number in the product. However, since we have excluded 2 (the only even prime), we need to consider the unit digits of the other primes.

- The unit digits of the primes greater than 2 are:
- 3 (unit digit 3)
- 5 (unit digit 5)
- 7 (unit digit 7)
- 1, 3, 7, 9 (from other primes)

When we multiply any number with a unit digit of 5 by an even number, the result will have a unit digit of 0.

### Conclusion
Since the product includes the prime number 5, and we have multiple odd primes, the unit digit of the product of all these primes will be 0.

### Final Answer
The unit digit in the product of the set of prime numbers greater than 2 but less than 222 is **0**.

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