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 $3^{45}$ ?

Cevap: 3

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Unit Digit Cyclicity

Tahmini Süre:45s

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