Cyclicity • Topic 1 of 2

Cycle of Powers

Only the unit digit of the base decides the unit digit of any power, so 23^45 has the same last digit as 3^45. Sort the ten digits by how their unit digit behaves under multiplication. Digits 0, 1, 5 and 6 are fixed points — they reproduce themselves at every power (period 1), so 6^anything ends in 6. Digits 4 and 9 toggle between two values (period 2): 4 gives 4, 6, 4, 6… and 9 gives 9, 1, 9, 1…, so an even power of 9 ends in 1 and an odd power in 9. The four "hard" digits 2, 3, 7, 8 each run through a 4-long cycle: 2→(2,4,8,6), 3→(3,9,7,1), 7→(7,9,3,1), 8→(8,4,2,6). A neat CAT shortcut: 7 and 3 share the same set {1,3,7,9}, and 2 and 8 share {2,4,6,8}. Learn these five mini-cycles cold and most unit-digit questions become instant.

✅ Solved examples

1. Find the unit digit of 6^59.
6 has period 1 — every power of 6 ends in 6. So the unit digit is 6.
2. Find the unit digit of 4^27.
4 has period 2: odd power → 4, even power → 6. 27 is odd, so the unit digit is 4.
3. Find the unit digit of 9^88.
9 has period 2: even power → 1, odd power → 9. 88 is even, so the unit digit is 1.
4. Find the unit digit of 2^10.
Cycle of 2 is (2,4,8,6), period 4. 10 mod 4 = 2, so take the 2nd term ⇒ 4.

✏️ Practice — try these, take hints as needed

1. Find the unit digit of 5^123.
Look only at the base digit 5.
5 has period 1.
Every power of 5 ends the same way.
5
2. Find the unit digit of 4^50.
4 has period 2 (4, 6).
Even power → 6, odd → 4.
50 is even.
6
3. Find the unit digit of 9^45.
9 has period 2 (9, 1).
Odd power → 9, even → 1.
45 is odd.
9
4. Find the unit digit of 11^200.
Only the last digit of 11 matters.
1 has period 1.
Every power of a number ending in 1 ends in 1.
1
5. Find the unit digit of 16^99.
Last digit of 16 is 6.
6 has period 1.
6 reproduces itself.
6

📝 Topic test — 8 questions

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