Concept 16 of 19Advanced
Watch on YouTubeVideoUnits-digit cyclicity (powers of a number)
The units digit of successive powers repeats in a short cycle:
- 2ⁿ: 2, 4, 8, 6, 2, 4, 8, 6, … (cycle length 4)
- 3ⁿ: 3, 9, 7, 1, 3, 9, 7, 1, … (cycle 4)
- 4ⁿ: 4, 6, 4, 6, … (cycle 2)
- 5ⁿ: always 5
- 6ⁿ: always 6
- 7ⁿ: 7, 9, 3, 1, 7, 9, 3, 1, … (cycle 4)
- 8ⁿ: 8, 4, 2, 6, 8, 4, 2, 6, … (cycle 4)
- 9ⁿ: 9, 1, 9, 1, … (cycle 2)
To find the units digit of bⁿ, figure out where n lands in the cycle: take n mod (cycle length).
Example
Units digit of 3¹⁰⁰?
Cycle of 3 is 4: (3, 9, 7, 1).
100 mod 4 = 0 → last in cycle → 1.
Cycle of 3 is 4: (3, 9, 7, 1).
100 mod 4 = 0 → last in cycle → 1.
💡 Tip:All non-zero units digits have cycles of 1, 2, or 4. No power of any single digit has a cycle of 3.
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5 questions to check what you just read.
0 / 5
- Q1.Units digit of 2⁸?
- Q2.Units digit of 3¹⁰?
- Q3.Units digit of 7¹⁰⁰?
- Q4.Units digit of 4⁵⁰?
- Q5.Units digit of 5²⁰⁰⁰?