If 3 is raised to the 59th power, i.e. 359, what is the units digit? The units digit is the digit in the ones place. Explain the method used to find your answer.
31=3
32=9
33=27
34=81
35=243
...
So we conclude that 3 raised to any power divisible by 4 ends with a 1.
Similarly, 3 raised to any power "divisible by 4 minus one", such as 3, 7, 11, ... ends with a 7.
Can you figure out the answer for 3^59 (noting that 59=4*15-1)?
To find the units digit of 3 raised to the 59th power (3^59), you can follow these steps:
Step 1: Start by listing the units digits of powers of 3:
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
3^5 = 243
3^6 = 729
...
Step 2: Observe the pattern in the units digits:
Notice that the units digits repeat every 4 powers. So, the units digits of the powers 3^1, 3^5, 3^9, 3^13, and so on, are all 3. Similarly, the units digits of 3^2, 3^6, 3^10, 3^14, and so on, are all 9. The pattern continues for the powers of 3^3 and 3^4 as well.
Step 3: Determine the position of 3^59 in the pattern:
Since the pattern repeats every 4 powers, and 59 is not a multiple of 4, we need to find the position of 3^59 in the pattern.
3^59 is the same as (3^4)^14 * 3^3, because 59 divided by 4 leaves a remainder of 3.
Step 4: Identify the units digit:
Now that we know 3^59 can be expressed as (3^4)^14 * 3^3, we can find the units digit of 3^59 by looking at the units digit of (3^4)^14, which is 1, and the units digit of 3^3, which is 7.
Therefore, the units digit of 3^59 is the units digit of (1 * 7), which is 7.
So, the units digit of 3^59 is 7.
To find the units digit of a number raised to a power, we need to observe a pattern.
First, let's understand the pattern of units digits when we raise 3 to consecutive powers:
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
3^5 = 243
As you can see, the units digits repeat in a cycle of 4: 3, 9, 7, 1.
Now, let's determine which number from the pattern corresponds to 3^59.
To find the position of 59 in the cycle, we divide 59 by 4: 59 ÷ 4 = 14 remainder 3.
Since 3 is the remainder, the units digit of 3^59 is the third number in the pattern, which is 7.
Therefore, the units digit of 3^59 is 7.