Find the unit digit of the product of all the elements of the set which consists all the prime numbers greater than 2 but less than 222.

Answer

To find the unit digit of the product of all prime numbers greater than 2 and less than 222, we can 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: Calculate the unit digit of the product
To find the unit digit of the product of these primes, we can focus on the unit digits of the individual primes:
- The unit digits of the primes are: 3, 5, 7, 1, 3, 7, 9, 3, 9, 1, 7, 1, 3, 7, 3, 9, 1, 7, 1, 3, 9, 3, 9, 7, 1, 3, 7, 9, 3, 1, 7, 9, 1, 5, 7, 3, 3, 7, 3, 9, 1, 3, 3, 7, 9, 1, 3.

### Step 3: Notice the presence of the digit 5
Among these primes, we have the prime number 5. When we multiply any number by 5, the unit digit of the product will be determined by the unit digit of the other number multiplied by 5.

### Step 4: Consider the effect of other primes
The unit digit of the product of any number with a unit digit of 5 will always end with a 5, regardless of the other digits involved in the multiplication.

### Conclusion
Since we have included the prime number 5 in our product, the unit digit of the product of all prime numbers greater than 2 and less than 222 will be **5**.