A number whose cube ends with the digit 7 must itself end with the digit 3.
Understanding Numbers Whose Cubes End in 7
When you cube an integer, the last digit of the resulting cube is determined solely by the last digit of the original number. For a perfect cube to have 7 as its unit digit, its base number must consistently have 3 as its unit digit.
Let's look at some examples:
- 3 cubed: $3 \times 3 \times 3 = 27$. The base number 3 ends in 3, and its cube, 27, ends in 7.
- 13 cubed: $13 \times 13 \times 13 = 2197$. The base number 13 ends in 3, and its cube, 2197, ends in 7.
- 23 cubed: $23 \times 23 \times 23 = 12167$. The base number 23 ends in 3, and its cube, 12167, ends in 7.
This pattern is consistent: any integer ending in 3, when cubed, will produce a result ending in 7. These numbers are examples of perfect cubes.
What if the Original Number Itself Ends in 7?
It's important to distinguish this from the reverse scenario. If an original number ends with the digit 7, its cube will not end in 7. Instead, the cube of a number ending in 7 will end in the digit 3. This is a common point of distinction when analyzing the unit digits of cubes.
Consider these examples:
- 7 cubed: $7 \times 7 \times 7 = 343$. The base number 7 ends in 7, and its cube, 343, ends in 3.
- 17 cubed: $17 \times 17 \times 17 = 4913$. The base number 17 ends in 7, and its cube, 4913, ends in 3.
- 27 cubed: $27 \times 27 \times 27 = 19683$. The base number 27 ends in 7, and its cube, 19683, ends in 3.
Understanding the Unit Digit Pattern of Cubes
The unit digit of a cube follows a predictable pattern based on the unit digit of its base number. This table summarizes the relationship:
Unit Digit of Base Number | Unit Digit of Cube |
---|---|
0 | 0 |
1 | 1 |
2 | 8 |
3 | 7 |
4 | 4 |
5 | 5 |
6 | 6 |
7 | 3 |
8 | 2 |
9 | 9 |
As illustrated, the digits 0, 1, 4, 5, 6, and 9 retain their unit digit when cubed. The pairs (2, 8) and (3, 7) show an inverse relationship: if a number ends in 2, its cube ends in 8, and vice-versa; similarly, if a number ends in 3, its cube ends in 7, and vice-versa.