Determine the last digit of the product of all even natural numbers less than 100 that are not multiples of ten.

(M. Volfová)
The last digit of the product is determined exclusively by the last digits of the factors. Therefore, in solving the problem, we will consider only the last digits in the calculations. According to the problem, we multiply ten sets of four factors, and in each set, the factors end with the digits 2, 4, 6, and 8. The product $2 \cdot 4 \cdot 6 \cdot 8$ ends with the digit 4. To determine the last digit of the result of multiplying ten such sets, it is sufficient to multiply ten fours and, in the multiplication, track only the last digit:

The product $4 \cdot 4$ ends with the digit 6, so instead of multiplying ten fours, it is sufficient to multiply five sixes. The number 6 multiplied by itself again gives a number ending with the digit 6, and therefore the product of five sixes, and thus also of ten fours, ends with the digit 6. The sought digit is 6.

Another solution. Again, we consider only the last digits in the products. We are given ten factors ending with the digit 2, ten ending with the digit 4, ten ending with the digit 6, and ten ending with the digit 8. We will consider each set of ten factors in turn:

When multiplying $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$, we track only the digits in the units place and find that the result ends with the digit 4. Similarly, we find that the product $4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4$ ends with the digit 6, the product $6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6$ ends with the digit 6, and the product $8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8 \cdot 8$ ends with the digit 4. The last digit of the sought product will be determined by multiplying the digits just found, i.e., $4 \cdot 6 \cdot 6 \cdot 4$. This product ends with the digit 6, so the sought digit is 6.

Suggested evaluation. 1 point for realizing that it is sufficient to consider only the last digits of the factors; 4 points for intermediate results and their justification (e.g., 2 points for the last digit of the product $2 \cdot 4 \cdot 6 \cdot 8$ and 2 points for the last digit of the power $4^{10}$); 1 point for the sought digit 6.