The image you provided shows a table labeled "Find the unit digit." Here's how we can determine the unit digit for each entry in the table:

Explanation:

To find the unit digit of a number raised to a power, observe the cyclic pattern in the unit digits of the powers of the base number.

For example:

• The unit digit of $2^1$ is 2.
• The unit digit of $2^2$ is 4.
• The unit digit of $2^3$ is 8.
• The unit digit of $2^4$ is 6.

Notice that the unit digits repeat every 4 steps: 2, 4, 8, 6.

So, to find the unit digit of $2^n$, you can find the remainder when $n$ is divided by 4, and use that to determine the corresponding digit in the cycle.

Let's apply this logic to the table:

1. 3^12:

   • Cycle of unit digits: 3, 9, 7, 1.
   • $12 \mod 4 = 0$, which means the unit digit corresponds to the last digit in the cycle, which is 1.

2. 7^10:

   • Cycle of unit digits: 7, 9, 3, 1.
   • $10 \mod 4 = 2$, so the unit digit corresponds to the second digit in the cycle, which is 9.

3. 8^15:

   • Cycle of unit digits: 8, 4, 2, 6.
   • $15 \mod 4 = 3$, so the unit digit corresponds to the third digit in the cycle, which is 2.

4. 9^17:

   • Cycle of unit digits: 9, 1.
   • $17 \mod 2 = 1$, so the unit digit corresponds to the first digit in the cycle, which is 9.

5. 6^20:

   • The unit digit of any power of 6 is always 6.

Final Answer:

• The unit digit of $3^{12}$ is 1.
• The unit digit of $7^{10}$ is 9.
• The unit digit of $8^{15}$ is 2.
• The unit digit of $9^{17}$ is 9.
• The unit digit of $6^{20}$ is 6.

Would you like more details on this topic or have any other questions?

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Related Questions:

1. How can you determine the unit digit for $5^{23}$?
2. What is the cyclic pattern of unit digits for the number 4?
3. How does the remainder when divided by 4 help in determining the unit digit of a number?
4. Can the pattern of unit digits for any base number ever be non-repetitive?
5. What is the significance of unit digits in modular arithmetic?

Tip:

Always look for a repeating cycle in the unit digits when dealing with powers to simplify the process of finding the last digit.