Definition: Cyclicity and unit digits refer to the mathematical study of the recurring patterns in the last digit of numbers raised to a specific power. By identifying the cycle length (periodicity) of each digit from 0 to 9, aspirants can determine the last digit of extremely large powers without performing full calculations.
Understanding the Concept of Cyclicity
When you raise a number to a power, the last digit follows a predictable, repeating sequence. This phenomenon is known as cyclicity. For example, consider the powers of 2: 21=2, 22=4, 23=8, 24=16 (ends in 6), and 25=32 (ends in 2). The pattern 2, 4, 8, 6 repeats every four steps.
In competitive exams like the UPSC CSAT, you are often asked to find the unit digit of complex expressions like 7253. Instead of calculating the value, you determine the cycle length of 7 and divide the exponent by that cycle length. The remainder tells you exactly where in the cycle the final digit falls.
Categorizing the Digits (0-9)
To simplify calculations, we group digits based on their specific cycle lengths. This categorization is a critical shortcut for saving time during the Quantitative Aptitude section of the exam.
- Cycle length 1: The digits 0, 1, 5, and 6 always result in the same unit digit regardless of the power (e.g., any number ending in 5 raised to any power will always end in 5).
- Cycle length 2: The digits 4 and 9 follow a shorter pattern. For 4: odd powers end in 4, even powers end in 6. For 9: odd powers end in 9, even powers end in 1.
- Cycle length 4: The digits 2, 3, 7, and 8 have a repeating cycle of four numbers. For these, you must divide the exponent by 4. If the remainder is 1, 2, or 3, the result corresponds to the 1st, 2nd, or 3rd position in the cycle; if the remainder is 0, it corresponds to the 4th position.
Applying the Remainder Rule
The most common mistake students make is misinterpreting the remainder 0. If you divide an exponent by 4 and get a remainder of 0, it does not mean the answer is the 0th power; it means the number is at the end of its cycle (the 4th power).
Rule: For any base ending in 2, 3, 7, or 8, if the exponent is n, calculate n mod 4. If the remainder is 0, use the unit digit of the number raised to the power of 4.
For example, to find the unit digit of 720: 20 divided by 4 leaves a remainder of 0. Therefore, we look at the 4th power of 7 (74 = 2401). The unit digit is 1.
Important Facts and Formulas
| Unit Digit (Base) | Cycle Length | Pattern |
|---|---|---|
| 0, 1, 5, 6 | 1 | Constant |
| 4 | 2 | 4, 6 |
| 9 | 2 | 9, 1 |
| 2, 3, 7, 8 | 4 | Varies (2,4,8,6 / 3,9,7,1 / 7,9,3,1 / 8,4,2,6) |
Previous Year Question Hints
Aspirants often encounter questions asking for the unit digit of a sum of powers, such as (123)123 + (456)456. To solve this:
- Solve the first part: 3123. Divide 123 by 4, remainder is 3. 33 = 27, so unit digit is 7.
- Solve the second part: 6456. Since any power of 6 ends in 6, the unit digit is 6.
- Add the results: 7 + 6 = 13. The final unit digit is 3.
Always watch out for negative signs or multiplication between terms, as the logic remains the same, but the final arithmetic step changes.
Quick Revision Summary
- Digits 0, 1, 5, 6 remain unchanged regardless of the exponent.
- Digits 4 and 9 alternate every two powers.
- Digits 2, 3, 7, 8 repeat every four powers.
- Always divide the exponent by the cycle length (4 for most cases).
- A remainder of 0 indicates the 4th power in the cycle.
- Focus on the unit digit of the base only; ignore all other digits to the left.
- Practice with mixed operations (addition, subtraction, multiplication) to build speed.
