units digit of the sum（Units Digit of the Sum）
神人无敌Units Digit of the Sum
The Problem
The units digit of a number is the digit in the one's place. For example, in the number 123, the units digit is 3. Now, consider the following problem:
Find the units digit of the sum (3^3 + 7^7 + 5^5 + 9^9 + 2^2).
The Approach
Let's break down the problem into smaller sub-problems. We need to find the units digit of each of the five numbers and then add them to find the units digit of the sum. Let's look at each number separately:
- 3^3: The last digit of 3^3 is 7. We can verify this by computing 3^3 = 27 and noticing that the units digit is indeed 7.
- 7^7: The last digit of 7^7 is 3. We can see that the units digit repeats every four powers of 7. That is, 7^1 has units digit 7, 7^2 has units digit 9, 7^3 has units digit 3, and 7^4 has units digit 1. Therefore, the units digit of 7^7 is the same as the units digit of 7^3, which is 3.
- 5^5: The last digit of 5^5 is 5. We can see that the units digit of 5^n is 5 for all positive integers n.
- 9^9: The last digit of 9^9 is 9. We can see that the units digit repeats every two powers of 9. That is, 9^1 has units digit 9, 9^2 has units digit 1, 9^3 has units digit 9, and so on. Therefore, the units digit of 9^9 is the same as the units digit of 9^1, which is 9.
- 2^2: The last digit of 2^2 is 4. We can verify this by computing 2^2 = 4 and noticing that the units digit is indeed 4.
Now, let's add the units digits of the five numbers to find the units digit of the sum:
7 + 3 + 5 + 9 + 4 = 28
Therefore, the units digit of the sum (3^3 + 7^7 + 5^5 + 9^9 + 2^2) is 8.
The General Method
What we have just done is an example of a general method to find the units digit of the sum of several numbers. The method can be summarized as follows:
- Find the units digit of each number.
- Add the units digits.
- If the result has more than one digit, add the digits again until only one digit remains.
Let's apply this method to a more general problem:
Find the units digit of the sum (a^n + b^n + c^n + ... + x^n + y^n + z^n), where n is a positive integer and a, b, c, ..., x, y, z are single-digit integers.
We can find the units digit of each number using the following table:
| a | a^2 | a^3 | a^4 | a^5 | a^6 | a^7 | a^8 | a^9 |
|---|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 4 | 8 | 6 | 2 | 4 | 8 | 6 | 2 |
| 3 | 9 | 7 | 1 | 3 | 9 | 7 | 1 | 3 |
| 4 | 6 | 4 | 6 | 4 | 6 | 4 | 6 | 4 |
| 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 |
| 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
| 7 | 9 | 3 | 1 | 7 | 9 | 3 | 1 | 7 |
| 8 | 4 | 2 | 6 | 8 | 4 | 2 | 6 | 8 |
| 9 | 1 | 9 | 1 | 9 | 1 | 9 | 1 | 9 |
For example, if a = 2 and n = 7, then the units digit of a^n is the units digit of 2^7, which is 8.
After finding the units digit of each number, we can add them and repeat the process until only one digit remains.
The Conclusion
As we have seen, it is possible to find the units digit of the sum of several numbers by finding the units digit of each number and adding them. The method can be generalized to calculate the units digit of the sum of any number of single-digit terms raised to any positive integer power. This technique is often used in mathematics competitions and can save a lot of time and effort if applied correctly.