Study Guide · Chapter 9
Cyclicity Table (unit digit of baseⁿ, n = 1, 2, 3, 4, …)
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| Base digit | Cycle of unit digits | Cycle length |
|---|---|---|
| 0 | 0, 0, 0, 0, … | 1 |
| 1 | 1, 1, 1, 1, … | 1 |
| 2 | 2, 4, 8, 6, 2, 4, 8, 6, … | 4 |
| 3 | 3, 9, 7, 1, 3, 9, 7, 1, … | 4 |
| 4 | 4, 6, 4, 6, … | 2 |
| 5 | 5, 5, 5, 5, … | 1 |
| 6 | 6, 6, 6, 6, … | 1 |
| 7 | 7, 9, 3, 1, 7, 9, 3, 1, … | 4 |
| 8 | 8, 4, 2, 6, 8, 4, 2, 6, … | 4 |
| 9 | 9, 1, 9, 1, … | 2 |
Method: 1. Note the unit digit of the base. 2. Find the cycle length for that digit (1, 2, or 4 from the table above). 3. Divide the exponent by the cycle length. If the remainder is r ≠ 0, the answer is the r-th term of the cycle. If the remainder is 0, the answer is the last term of the cycle (i.e., treat remainder as equal to the cycle length).
Why it works: Multiplying repeatedly by the same digit modulo 10 must eventually repeat, because there are only 10 possible remainders (0–9). The cycles above are simply this repetition worked out for each digit.
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