Remainders & Modular Reasoning • Topic 2 of 4

Cyclicity of Units Digits

The units digit of a power repeats in a short cycle, so you never compute the whole number. The cycles all have length 1, 2 or 4: digits 0, 1, 5 and 6 keep the same units digit at every power; 4 and 9 alternate in a cycle of 2 (4→6→4…, 9→1→9…); and 2, 3, 7, 8 run in cycles of length 4 (2: 2,4,8,6; 3: 3,9,7,1; 7: 7,9,3,1; 8: 8,4,2,6). To find the units digit of a power, take the exponent modulo the cycle length and read the matching position. For 7⁴⁵, the cycle length is 4 and 45 = 4×11 + 1, so the units digit matches 7¹ = 7. This is one of the few GRE topics where a memorised table gives an instant answer the calculator cannot reach for large exponents.

✅ Solved examples

1. What is the units digit of 2¹⁰?
Cycle of 2 is 2,4,8,6 (length 4). 10 = 4×2 + 2, so use position 2 in the cycle → 4. (Indeed 2¹⁰ = 1024.)
2. What is the units digit of 3¹⁰⁰?
Cycle of 3 is 3,9,7,1 (length 4). 100 = 4×25 + 0, i.e. a multiple of 4 → the last position, units digit 1.
3. What is the units digit of 7⁴⁵?
Cycle of 7 is 7,9,3,1 (length 4). 45 = 4×11 + 1 → position 1 → units digit 7.
4. Quantitative Comparison. Quantity A: the units digit of 4²⁰. Quantity B: the units digit of 9²⁰. Which is greater?
4 cycles 4,6: even exponent → 6. 9 cycles 9,1: even exponent → 1. Quantity A (6) > Quantity B (1): Quantity A is greater.

✏️ Practice — try these, take hints as needed

1. Units digit of 2²³?
Cycle 2,4,8,6.
23 mod 4 = 3.
Third in the cycle.
8
2. Units digit of 8¹²?
Cycle 8,4,2,6.
12 mod 4 = 0.
Last position.
6
3. Units digit of 9³¹?
Cycle 9,1.
Odd exponent.
First position.
9
4. Units digit of 6⁵⁰?
6 is stable.
Every power ends in 6.
No cycle needed.
6
5. Units digit of 3⁴⁷?
Cycle 3,9,7,1.
47 mod 4 = 3.
Third position.
7

📝 Topic test — 8 questions

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