Home › Number theory

How to find the last digit of a large power

The rule: find the cycle, then reduce the exponent

Only the last digit of the base affects the last digit of a power. List successive last digits until they repeat, divide the exponent by the cycle length, and use the remainder as the position in the cycle. A remainder of $0$ means the last position, not the first.

Problem

What is the last digit in the decimal notation of $3^{144}$ ? Choose from $1$ , $3$ , $7$ and $9$ .

Answer

The last digit of

3^{144}

1

Checked by the four-step cycle modulo 10 and by 144 mod 4 = 0.

Step-by-step solution

Find the repeating pattern

Calculate only the last digit of the first few powers:

3^{1} \equiv 3, 3^{2} \equiv 9, 3^{3} \equiv 7, 3^{4} \equiv 1 (mod 10)

Multiplying by $3$ once more returns to a last digit of $3$ , so the cycle $3, 9, 7, 1$ repeats every four powers.

Locate the exponent in the cycle

Divide $144$ by the cycle length:

144 = 4 \cdot 36 + 0

The remainder is $0$ , so the exponent lands on the fourth item in the cycle. That item is $1$ . Therefore

3^{144} \equiv 1 (mod 10)

How to check yourself

For a four-item cycle, remainders $1$ , $2$ , $3$ and $0$ correspond to positions $1$ , $2$ , $3$ and $4$ . Treating remainder $0$ as position $0$ is the usual source of a wrong answer.

Add to Chrome - solve your own

Screenshot any problem on your screen and get the steps. Chrome on a computer; free.