What is the digit in the units place of the number (1829)⁴²?

Given:

What is the digit in the units place of the number (1829)⁴²?

Formula used:

To find the unit digit of a number raised to a power, we use the cyclicity of the unit digits in powers of the base number.

Calculation:

The unit digit of 1829 is 9.

Unit digits of powers of 9 follow a cyclic pattern:

9¹ = 9 → Unit digit = 9

9² = 81 → Unit digit = 1

9³ = 729 → Unit digit = 9

9⁴ = 6561 → Unit digit = 1

Cyclic pattern: 9, 1

Now, divide the exponent (42) by the length of the cycle (2) to find the remainder:

⇒ 42 ÷ 2 = 21 remainder 0

When the remainder is 0, the unit digit corresponds to the last number in the cycle.

From the cycle (9, 1), the last number is 1.

∴ The digit in the units place of (1829)⁴² is 1.