What is the units digit of 7^{43}? — GMAT

What is the units digit of $7^{43}$ ?

Answer: 3

Answer

The units digit of

7^{43}

is 3.

The units digits of powers of 7 repeat in a pattern of four terms: 7, 9, 3, 1. Dividing the exponent 43 by 4 yields a remainder of 3. The 3rd term in the repeating pattern is 3, so the units digit of

7^{43}

is 3.

Step-by-Step Solution

Determine the cyclicity pattern of the units digit of powers of 7.

The units digits for

7^1, 7^2, 7^3, 7^4, \dots

follow the repeating sequence 7, 9, 3, 1 with a cycle length of 4.

Units digits of positive integer powers follow a periodic pattern.

Divide the exponent 43 by the pattern cycle length of 4.

43 \div 4 = 10

remainder 3.

The remainder determines which term in the repeating sequence gives the units digit.

Find the units digit corresponding to the 3rd term in the cyclicity sequence.

The 3rd digit in the sequence (7, 9, 3, 1) is 3.

A remainder of 3 corresponds to the 3rd power in the cycle,

7^3

Key Concept

Units Digit Cyclicity

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic