How it works
The units digit of a product depends only on the units digits of the factors. So the units digits of powers repeat:
| Base ends in | Cycle of units digits |
|---|---|
| 2 | 2, 4, 8, 6 |
| 3 | 3, 9, 7, 1 |
| 4 | 4, 6 |
| 7 | 7, 9, 3, 1 |
| 8 | 8, 4, 2, 6 |
| 9 | 9, 1 |
| 0, 1, 5, 6 | always the same digit |
Every cycle has length 1, 2 or 4, so divide the exponent by 4 and use the remainder (a remainder of 0 means the last number in the cycle). For sums, find each units digit separately, add them and take the last digit.
Units Digits And Cycles practice problems with solutions
These problems come from Mathyessy's question bank. Try each one before opening the solution.
What is the units digit of 7 to the power 2026?
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Answer: 9
Powers of 7 end in 7, 9, 3, 1, repeating every 4. 2026 leaves remainder 2 when divided by 4, so the units digit matches 7², which is 9.
What is the units digit of 2 to the power 25?
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Answer: 2
Powers of 2 end in 2, 4, 8, 6. 25 leaves remainder 1 when divided by 4, so the units digit is 2.
What is the units digit of 7 to the power 77 plus 3 to the power 77?
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Answer: 0
77 leaves remainder 1 when divided by 4, so 7⁷⁷ ends in 7 and 3⁷⁷ ends in 3. 7 + 3 = 10 ends in 0.
What is the units digit of 2 to the power 100 plus 3 to the power 100 plus 7 to the power 100?
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Answer: 8
100 is a multiple of 4, so take the last digit of each cycle: 6, 1 and 1. 6 + 1 + 1 = 8.
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