Number System Fundamentals • Topic 4 of 7

Unit Digit

The units (last) digit of a power repeats in a short cycle. Powers of 2 cycle 2, 4, 8, 6; powers of 3 cycle 3, 9, 7, 1; powers of 7 cycle 7, 9, 3, 1; powers of 8 cycle 8, 4, 2, 6 — each of length 4. The digits 0, 1, 5 and 6 never change, while 4 and 9 alternate with a cycle of length 2. To find the units digit of a power, divide the exponent by the cycle length and read off the matching term. The units digit of a product depends only on the units digits of the factors.

✅ Solved examples

1. What is the units digit of 9⁸⁸?
Powers of 9 alternate 9, 1 (cycle length 2): odd powers end in 9, even powers end in 1. Since 88 is even, the units digit is 1.
2. Find the units digit of 2⁵⁰.
Powers of 2 cycle 2, 4, 8, 6 (length 4). 50 ÷ 4 leaves remainder 2, so take the 2nd term of the cycle: 4.
3. What is the units digit of 7¹²³?
Powers of 7 cycle 7, 9, 3, 1 (length 4). 123 ÷ 4 leaves remainder 3, so take the 3rd term: 3.
4. Find the units digit of 13 × 27 × 35.
Only the units digits matter: 3 × 7 × 5. First 3 × 7 = 21 (units 1), then 1 × 5 = 5. The units digit is 5.

✏️ Practice — try these, take hints as needed

1. What is the units digit of 3⁴⁰?
Powers of 3 cycle 3, 9, 7, 1 (length 4).
40 ÷ 4 leaves remainder 0, meaning the last term of the cycle.
The 4th term is 1.
1.
2. Find the units digit of 4¹⁷.
Powers of 4 alternate 4, 6 (cycle length 2).
Odd power → 4, even power → 6.
17 is odd.
4.
3. What is the units digit of 8²⁵?
Powers of 8 cycle 8, 4, 2, 6 (length 4).
25 ÷ 4 leaves remainder 1.
Take the 1st term of the cycle.
8.
4. Find the units digit of 17⁶.
Only the units digit 7 matters; use the cycle 7, 9, 3, 1.
6 ÷ 4 leaves remainder 2.
Take the 2nd term.
9.
5. What is the units digit of 2³³ × 3³³?
Find each units digit separately. 2: 33 ÷ 4 r1 → 2. 3: 33 ÷ 4 r1 → 3.
Multiply the two units digits.
2 × 3 = 6.
6.

📝 Topic test — 8 questions

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