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

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

A
2
B
4
C
6
8Cevap
E
0

Cevap

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.

Adım Adım Çözüm

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

Anahtar Kavram

Units Digit Cyclicity

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