﻿ Find the unit digits of the large powers problems.

#### Question

Find the unit digits of the large powers problems.

What is the units digit of 24 to the power of 2023?

The units digit of 13^{34}

Find the units digit in the expansion of 163 raised to 61?

The units digit in the expansion of 2 raised to the power of 91?

The last digit in 6 of power 2019 is?

What is the unit digit in 57^{2020}?

Collected in the board: Number Theory

Steven Zheng posted 1 week ago

When a number is raised to some power, the units digit is also raised to the same power. the units digit of the final number rotates among certain numbers matter what power it is raised to.

Find the units digit of 24 to the power of 2023 is to find the units digit of 24^{2023}. The units unit of the base 24 is 4.

If 4 is raised to the power 1, the units is 4.

If 4 is raised to the power 2, the units digit is 6.

And then repeat as 4 is raised to large powers in the way if the power is odd, the units digit is 4 and if the power is even, the units digit is 6.

Since the exponent 2023 is odd, the units digit of 24^{2023} is 4

The units digit of 13^{34}

For the number 3 raised to powers, its units digit changes as follow

3^1 = 3

3^2 = 9

3^3 = 27

3^4 =81

And then repeat no matter what power is raised to

The exponent 34, when divided by 4 leaves reminder 2. Then The units digit of 13^{34} is given as 9.

Find the units digit in the expansion of 163 raised to 61 is to find the units digit of 163^{61}

The exponent is 61, which when divided by 4 gives a remainder 1.

Therefore, the units digit of 163^{61} is 3

Here, we have to find the units digit of 2^{91}

For the number 2 raised to powers, its units digit changes as follow

2^1 = 2

2^2 = 4

2^3 = 8

2^4 = 16

And then, repeat no matter what power is raised to.

Since the exponent 91 when divided by 4 leaves remainder 1. Hence, the units digit of 2 to the power of 91 is given as 2.

The last digit of 6 of power 2019

For the number 6, no matter what power is raised to, the units digit of the final number is always 6. Hence, the last digit of 6 of power 2019 is 6.

What is the unit digit in 57^{2020}?

For the number 7 raised to powers, its units digit could be 7, 9,3,1

7^1 = 7

7^2 = 49

7^3 = 343

7^4 = 2401

The power 2020 when divided by 4 leaves remainder 0. Therefore, the units digit of 57 to the power of 2020 is given as 1.

Steven Zheng posted 1 week ago

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