Finding the Units Digit of $12687^{155}$

To find the units digit of $12687^{155}$, we only need to consider the units digit of the base number, which is 7.

Analyzing Powers of 7

Let's examine the pattern of the units digits of the powers of 7:

• $7^1$ has a units digit of 7.
• $7^2 = 49$ has a units digit of 9.
• $7^3 = 343$ has a units digit of 3.
• $7^4 = 2401$ has a units digit of 1.
• $7^5 = 16807$ has a units digit of 7.

The pattern of the units digits of powers of 7 is (7, 9, 3, 1). This cycle repeats every 4 powers.

Determining the Position in the Cycle

To find the units digit of $7^{155}$, we need to determine where in this 4-digit cycle the 155th power falls. We do this by finding the remainder when the exponent, 155, is divided by the cycle length, 4.

Calculation:

The remainder is 3.

Final Units Digit

A remainder of 3 means the units digit corresponds to the 3rd number in the cycle (7, 9, 3, 1).

The 3rd digit in the cycle is 3.

Therefore, the units digit of $12687^{155}$ is 3.