Units Digit of 12502¹⁴⁹ Calculation

To find the units digit of a large power like $12502^{149}$, we only need to focus on the units digit of the base number.

The base number is 12502. Its units digit is 2. Thus, the problem reduces to finding the units digit of $2^{149}$.

Units Digit Pattern for Powers of 2

Let's observe the pattern of the units digit for the first few powers of 2:

• $2^1 = 2$
• $2^2 = 4$
• $2^3 = 8$
• $2^4 = 16 \implies 6$
• $2^5 = 32 \implies 2$

The pattern of the units digit is (2, 4, 8, 6), which repeats every 4 powers. The cycle length is 4.

Finding the Position in the Cycle

To determine the units digit of $2^{149}$, we find the remainder when the exponent (149) is divided by the cycle length (4).

Calculate the remainder:

$149 \pmod{4}$

$149 = 4 \times 37 + 1$

The remainder is 1.

Determining the Final Units Digit

A remainder of 1 corresponds to the first element in the cycle (2, 4, 8, 6).

Therefore, the units digit of $2^{149}$ is 2.

Consequently, the units digit of $12502^{149}$ is 2.