What is the units digit of 8^{21}? — GMAT

What is the units digit of $8^{21}$ ?

The units digits of integer powers of 8 follow a repeating pattern of length 4: 8, 4, 2, 6. Dividing the exponent 21 by 4 yields

21 = 4 \times 5 + 1

, giving a remainder of 1. A remainder of 1 means the units digit is the first element of the cycle, which is 8.

Determine the units digit pattern for powers of 8.

The sequence of units digits for

8^1, 8^2, 8^3, 8^4, \dots

8, 4, 2, 6, 8, 4, 2, 6, \dots

, which repeats every 4 powers.

Units digits of positive integer powers follow a periodic cyclic pattern.

Find the remainder when the exponent 21 is divided by the cycle length 4.

21 \div 4 = 5

with a remainder of 1.

The remainder determines the position of the units digit within the 4-step cycle.

Identify the units digit corresponding to a remainder of 1.

The first number in the pattern

8, 4, 2, 6

is 8.

A remainder of 1 points to the 1st term of the repeating sequence.

Units Digit Cyclicity

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