What is the unit digit in 653^723?

To find the unit digit of 653^723, we only need to consider the unit digit of the base number, which is 3.

The powers of 3 follow a pattern in their unit digits:

3^1 = 3 3^2 = 9 3^3 = 27 3^4 = 81 3^5 = 243

As you can see, the unit digit repeats every 4 powers: 3, 9, 7, 1.

To find out which number in this cycle is the unit digit of 653^723, we divide the exponent by 4.

723 ÷ 4 = 180 remainder 3

The remainder tells us which number in the cycle is the unit digit. In this case, the remainder is 3, so the unit digit of 653^723 is the third number in the cycle, which is 7.

So, the unit digit of 653^723 is 7.