In basic arithmetic, the unit digit of a power of an integer repeats in a periodic pattern… — State PSC Exam

In basic arithmetic, the unit digit of a power of an integer repeats in a periodic pattern known as cyclicity. What is the unit digit of $3^{45}$ ?

Answer: 3

Answer

The unit digit of

3^{45}

is 3.

The base number 3 has a cyclicity pattern of length 4: (3, 9, 7, 1). When the exponent 45 is divided by 4, the quotient is 11 and the remainder is 1. A remainder of 1 indicates the unit digit is equal to the first element of the cyclic pattern, which is 3.

Step-by-Step Solution

Identify the unit digit cyclicity pattern for base 3.

The unit digits repeat every 4 powers in the pattern (3, 9, 7, 1).

Calculating sequential powers of 3 yields last digits of 3, 9, 7, 1, 3, 9, 7, 1, establishing a cycle length of 4.

Divide the exponent by the cycle period of 4.

45 \div 4 = 11

with a remainder of

1

The remainder indicates the exact position within the repeating cycle.

Find the unit digit using the remainder.

Since the remainder is 1, the unit digit is

3^1 = 3

The first number in the cyclicity sequence (3, 9, 7, 1) is 3.

Key Concept

Unit Digit Cyclicity

Estimated Time:45s

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