What is the units digit of 2^{35}? — GMAT

What is the units digit of $2^{35}$ ?

A
2
B
4
C
6
8Answer
E
0

Answer

The units digit of

2^{35}

is 8.

The units digit of powers of 2 repeats every 4 powers in the sequence 2, 4, 8, 6. Dividing the exponent 35 by 4 yields a quotient of 8 and a remainder of 3. A remainder of 3 corresponds to the third number in the repeating sequence, which is 8.

Step-by-Step Solution

Find the cyclicity pattern of the units digits for powers of 2.

The units digits of

2^1, 2^2, 2^3, 2^4, 2^5, \dots

are

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

, repeating in a cycle of length 4.

Units digits of powers follow a repeating periodic pattern determined by base arithmetic modulo 10.

Divide the exponent by the length of the cycle.

35 \div 4 = 8

with a remainder of

3

The remainder indicates how far into the 4-term repeating cycle the exponent 35 reaches.

Match the remainder to the corresponding term in the units digit cycle.

A remainder of 3 corresponds to the 3rd term in the cycle

(2, 4, 8, 6)

, which is 8.

The 3rd power in the repeating pattern gives a units digit of

2^3 = 8

Key Concept

Units Digit Cyclicity

Estimated Time:1m 0s

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