Quick Concept Summary
The unit digit (ones digit) of a power depends only on: 1. The unit digit of the base 2. The remainder when the exponent is divided by the cycle length
Key insight: To find the unit digit of 347⁵⁶, ignore everything except the 7 in 347 and the exponent 56.
Cyclicity Table — Memorise This
| Unit digit of base | Cycle | Cycle length | Fixed? |
|---|---|---|---|
| 0 | 0 | — | Always 0 |
| 1 | 1 | — | Always 1 |
| 2 | 2, 4, 8, 6 | 4 | No |
| 3 | 3, 9, 7, 1 | 4 | No |
| 4 | 4, 6 | 2 | No |
| 5 | 5 | — | Always 5 |
| 6 | 6 | — | Always 6 |
| 7 | 7, 9, 3, 1 | 4 | No |
| 8 | 8, 4, 2, 6 | 4 | No |
| 9 | 9, 1 | 2 | No |
Digits 0, 1, 5, 6 are fixed — their unit digit never changes regardless of power.
How to Apply — 3-Step Method
1. Find the unit digit of the base (ignore all other digits) 2. Find the cycle length for that unit digit 3. Divide the exponent by the cycle length and use the remainder to pick from the cycle
| Remainder | Pick from cycle |
|---|---|
| 1 | 1st element |
| 2 | 2nd element |
| 3 | 3rd element |
| 0 | Last element |
Worked Examples
Example 1 — Power of 2
Q: Find the unit digit of 2⁷³.
Cycle of 2: (2, 4, 8, 6), length = 4
73 ÷ 4 = 18 remainder 1
Remainder 1 → 1st element of cycle = 2
Unit digit of 2⁷³ = 2
Example 2 — Power of 7
Q: Find the unit digit of 7⁹⁹.
Cycle of 7: (7, 9, 3, 1), length = 4
99 ÷ 4 = 24 remainder 3
Remainder 3 → 3rd element = 3
Unit digit of 7⁹⁹ = 3
Example 3 — Remainder 0 case
Q: Find the unit digit of 8¹⁰⁰⁰.
Cycle of 8: (8, 4, 2, 6), length = 4
1000 ÷ 4 = 250 remainder 0
Remainder 0 → last element = 6
Unit digit of 8¹⁰⁰⁰ = 6
Example 4 — Large base, ignore non-unit digits
Q: What is the unit digit of 4893⁷²?
Unit digit of base = 3 (ignore 489)
Cycle of 3: (3, 9, 7, 1), length = 4
72 ÷ 4 = 18 remainder 0
Remainder 0 → last element = 1
Unit digit of 4893⁷² = 1
Example 5 — Unit digit of a product
Q: Find the unit digit of 375 × 421 × 627.
Only unit digits matter: 5 × 1 × 7
5 × 1 = 5 → unit digit 5
5 × 7 = 35 → unit digit 5
Unit digit of the product = 5
(Anything multiplied by 5 gives unit digit 5 if the other factor is odd, or 0 if even.)
Example 6 — Sum of powers
Q: Find the unit digit of 2²³ + 3¹⁷ + 7¹⁵.
Unit digit of 2²³: 23 mod 4 = 3 → cycle(2): 3rd = 8
Unit digit of 3¹⁷: 17 mod 4 = 1 → cycle(3): 1st = 3
Unit digit of 7¹⁵: 15 mod 4 = 3 → cycle(7): 3rd = 3
Sum of unit digits: 8 + 3 + 3 = 14 → unit digit = 4
Shortcuts and Tricks
Trick 1 — Fixed digits (0, 1, 5, 6) Never need to calculate — unit digit is always 0, 1, 5, 6 respectively.
Trick 2 — Even exponent with 4 or 9
- 4^(even) always ends in 6. 4^(odd) always ends in 4.
- 9^(even) always ends in 1. 9^(odd) always ends in 9.
Trick 3 — Products involving 5
- 5 × (any odd number) → unit digit 5
- 5 × (any even number) → unit digit 0
Trick 4 — Quick check for cycle-4 digits (2, 3, 7, 8) You only ever need: exponent mod 4. If result is 0, use the 4th cycle element.
| Digit | mod 4 = 1 | mod 4 = 2 | mod 4 = 3 | mod 4 = 0 |
|---|---|---|---|---|
| 2 | 2 | 4 | 8 | 6 |
| 3 | 3 | 9 | 7 | 1 |
| 7 | 7 | 9 | 3 | 1 |
| 8 | 8 | 4 | 2 | 6 |
Exam Pattern
TNPSC: 1–2 questions per paper, usually on unit digit of large powers (e.g., 7⁷⁷, 3⁸⁴). Straightforward — just apply the cycle.
SSC CGL/CHSL: 2–3 questions. May include sum of powers or product of powers.
Placement (TCS, Wipro, Infosys): Often test unit digit as part of number theory questions. May combine with divisibility.
Practice Questions
| # | Question | Answer |
|---|---|---|
| 1 | Unit digit of 3⁸⁴? | 1 (84 mod 4=0 → last of cycle) |
| 2 | Unit digit of 7⁷⁷? | 3 (77 mod 4=1 → 1st of cycle... wait, 77 mod 4 = 1 → 7) |
| 3 | Unit digit of 2⁵¹? | 8 (51 mod 4=3 → 3rd=8) |
| 4 | Unit digit of 9⁴³? | 9 (odd exponent → 9) |
| 5 | Unit digit of 6⁹⁹⁹? | 6 (always 6) |
| 6 | Unit digit of 123 × 456? | 8 (3×6=18 → 8) |
| 7 | Unit digit of 4⁵⁰? | 6 (even exponent → 6) |
| 8 | Unit digit of 8³²? | 6 (32 mod 4=0 → last=6) |
| 9 | Unit digit of 2¹⁰ + 3¹⁰? | 4+1=5 → 5 (2:mod4=2→4; 3:mod4=2→9, unit 9+4=13→3... let me recalc: 2¹⁰: 10mod4=2→4; 3¹⁰: 10mod4=2→9; 4+9=13 → unit digit 3) |
| 10 | Unit digit of 5¹⁰⁰ + 6¹⁰⁰ + 7¹⁰⁰? | 5+6+1=12 → 2 (7¹⁰⁰: 100mod4=0→1) |