What is the last digit of the number
4^2006
start listing the powers. You will see a pattern which makes it easy to solve this problem.
4^1= 4
4^2 = 16
4^3 = 64
4^4= 256
4^5 = 1024
You see a pattern?
4,6,4,6,4
every time the power is odd, it ends with 4
every time the power is even, it ends with 6
2006 is even, so the last digit is 6
To find the last digit of a number, you need to determine the pattern of the last digits as the number increases.
Let's analyze the pattern for the powers of 4. Start by listing a few powers of 4:
4^1 = 4
4^2 = 16
4^3 = 64
4^4 = 256
4^5 = 1024
4^6 = 4096
4^7 = 16384
From these examples, we can observe that the last digit is repeating in a cycle of 4, 6, 4, 6, etc. This means that for any power of 4, the last digits will follow this pattern: 4, 6, 4, 6, etc.
To determine the last digit of 4^2006, we need to find out where 2006 falls in this pattern.
Since the pattern repeats every 2 powers, we can divide 2006 by 2 to see how many complete cycles there are. 2006 divided by 2 is 1003, with a remainder of 0.
Since the remainder is 0, we are at the beginning of a new cycle, so the last digits will be 4.
Therefore, the last digit of 4^2006 is 4.