To find the unit digit of a cube, you only need to look at the unit digit of the original number, cube it, and then identify the unit digit of that result. This is because the higher-place digits of a number do not influence the unit digit of its cube.
The Simple Rule for Finding a Cube's Unit Digit
The most straightforward way to determine the unit digit of any number's cube is to:
- Identify the unit digit of the number you are cubing.
- Cube this single digit.
- The unit digit of this smaller cubed result will be the unit digit of the original number's cube.
For instance, if you want to find the unit digit of 12³, you would identify that the unit digit of 12 is 2. Then, you cube 2 (2³ = 8). The unit digit of 8 is 8, so the unit digit of 12³ is 8.
Unit Digits of Cubes (0-9) - The Pattern
The unit digit of a cube follows a consistent pattern based on the unit digit of the base number. Observing the cubes of single digits from 0 to 9 reveals this pattern:
| Unit Digit of Original Number | Cube of Unit Digit | Unit Digit of the Cube |
|---|---|---|
| 0 | 0³ = 0 | 0 |
| 1 | 1³ = 1 | 1 |
| 2 | 2³ = 8 | 8 |
| 3 | 3³ = 27 | 7 |
| 4 | 4³ = 64 | 4 |
| 5 | 5³ = 125 | 5 |
| 6 | 6³ = 216 | 6 |
| 7 | 7³ = 343 | 3 |
| 8 | 8³ = 512 | 2 |
| 9 | 9³ = 729 | 9 |
Key Observations:
- If the unit digit of the original number is 0, 1, 4, 5, 6, or 9, the unit digit of its cube will be the same digit.
- If the unit digit is 2, the cube's unit digit is 8.
- If the unit digit is 3, the cube's unit digit is 7.
- If the unit digit is 7, the cube's unit digit is 3.
- If the unit digit is 8, the cube's unit digit is 2.
Notice that 2 and 8 are complementary (2+8=10), and 3 and 7 are complementary (3+7=10) in terms of their cube's unit digits.
Practical Examples
Let's apply this rule with a few examples:
-
Example 1: Find the unit digit of 46³
- The unit digit of 46 is 6.
- Cube 6: 6³ = 216.
- The unit digit of 216 is 6.
- Therefore, the unit digit of 46³ is 6.
-
Example 2: Find the unit digit of 197³
- The unit digit of 197 is 7.
- Cube 7: 7³ = 343.
- The unit digit of 343 is 3.
- Therefore, the unit digit of 197³ is 3.
-
Example 3: Find the unit digit of 528³
- The unit digit of 528 is 8.
- Cube 8: 8³ = 512.
- The unit digit of 512 is 2.
- Therefore, the unit digit of 528³ is 2.
By following this simple method and understanding the pattern of unit digits, you can quickly determine the unit digit of any number's cube without calculating the entire cube.
