Use inductive reasoning to determine the units digit of the number 2^58
How do I go about finding the answer? I've been looking through my textbook and still can't fully understand
look at the units digit as the powers increase
n 2^n ends in
1 2
2 4
3 8
4 6
5 2
...
so 2^4n ends in 6
58 = 4*16 + 2, so 2^58 ends in 4
To determine the units digit of the number 2^58 using inductive reasoning, you can follow these steps:
Step 1: Note the pattern:
Start by finding the units digits of powers of 2. Observe how the units digit repeats in cycles. For example:
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 6
2^5 = 2
2^6 = 4
2^7 = 8
2^8 = 6
You can see that the units digit repeats after every four powers.
Step 2: Reduce the exponent to a multiple of 4:
Since the pattern repeats every four powers, we can simplify the exponent 58 by finding its remainder when divided by 4. In this case, 58 divided by 4 leaves a remainder of 2.
Step 3: Identify the units digit:
Knowing the remainder, we can determine that 2^58 will have the same units digit as 2^2. From the pattern in Step 1, we see that 2^2 is 4.
Hence, the units digit of the number 2^58 is 4.
To determine the units digit of a number, we need to consider the pattern or cycle of the units digits of its powers. Let's start by observing the units digits of powers of 2:
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
2^7 = 128
2^8 = 256
2^9 = 512
By examining these powers, we can notice a pattern: the units digit repeats after every 4 powers. That means that if we can find the remainder when dividing the exponent (58) by 4, we can determine which power's units digit corresponds to 2^58.
To find the remainder when dividing 58 by 4, divide 58 by 4 and look at the remainder: 58 ÷ 4 = 14 remainder 2. So, the remainder is 2.
Now, we need to find which power's units digit corresponds to 2 raised to the power of 2. In other words, we need to find the units digit of 2^2.
From the earlier observation, we know that:
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 6 (units digit repeated)
2^5 = 2 (units digit repeated)
2^6 = 4 (units digit repeated)
2^7 = 8 (units digit repeated)
2^8 = 6 (units digit repeated)
Since the units digit repeats every 4 powers, and we have a remainder of 2 when dividing 58 by 4, we know that the units digit of 2^58 will be the same as the units digit of 2^2. Therefore, the units digit of 2^58 is 4.
In summary, by using the pattern of the units digits of powers of 2, we determined that the units digit of 2^58 is 4.