a) Use a calculator to compute 2^161. Are you able to tell what the ones digit of this number is from the calculator’s display? Why or why not.
b) Determine the ones digit of 2^161. Explain your reasoning clearly.
If your calculator won't compute that high a number... why not start with 2^2, then 2^3, then 2^4 and so on until you see the pattern : )
2^161 ≈ 3E48 ... 3 x 10^48
a) no ... not enough digits on calculator
b) 2 , 4 , 8 , 16 , 32 , 64 , 128 , 256 , ...
the units digit repeats in a pattern of four numbers ... 2, 4, 8, 6
dividing 161 by 4 leaves a remainder of 1
... the 1st number in the repeating pattern is 2
... this is the units digit of 2^161
a) No, it is not possible to determine the ones digit of 2^161 just by looking at the calculator's display. The calculator can only display a limited number of digits, typically around 8 to 12, depending on the model. The number 2^161 is extremely large and cannot be accurately displayed on a calculator, as it would require significantly more digits.
b) To determine the ones digit of 2^161, we can analyze the patterns that occur with the powers of 2. The ones digit of 2^n repeats in a cycle of 4: 2, 4, 8, 6. This means that for every multiple of four, the ones digit will be the same.
For 161, we can divide it by 4 to find the closest multiple of 4 less than it. 161 divided by 4 is 40 with a remainder of 1, so the closest multiple of 4 is 40. This means that the ones digit of (2^4)^40 is the same as the ones digit of 2^40.
Since the ones digit of 2^4 cycle through 2, 4, 8, 6, and 40 is divisible by 4, the ones digit of 2^40 is 6. Therefore, the ones digit of 2^161 is also 6.
a) When using a calculator to compute 2^161, the calculator will typically display the result as a decimal number with multiple digits after the decimal point. The calculator does not directly show the ones digit of the result.
b) To determine the ones digit of 2^161, we can utilize the concept of cyclicity or periodicity in numbers. The ones digit of any power of 2 follows a repeating pattern. Let's observe some initial powers of 2:
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16 (ones digit is 6)
2^5 = 32 (ones digit is 2)
2^6 = 64 (ones digit is 4)
2^7 = 128 (ones digit is 8)
2^8 = 256 (ones digit is 6)
We notice that after every 4th power, the ones digit repeats in the pattern {2, 4, 8, 6}. Therefore, the ones digit of 2^161 will be the same as the ones digit of 2^1, which is 2.
Hence, the ones digit of 2^161 is 2.