Question

What is the units digit in the expansion of 2 to the power of 727 ?

8
4
2
6

Answer 1

Answer:

6

Step-by-step explanation:

6²= 36

6+1=7

36*2 = 72

72,7

727

Answer 2

Option A : The unit digit in the power of 2 to 727 will be 8.

Power 1 of 2  : $2^{1}$ = 2       [ Unit digit in 2 to power 1 is 2 ]

Power 2 of 2 : $2^{2}$ = 4       [ Unit digit in 2 to power 2 is 4 ]

Power 3 of 2 : $2^{3}$ = 8       [ Unit digit in 2 to power 3 is 8 ]

Power 4 of 2 : $2^{4}$ = 16     [ Unit digit in 2 to power 4 is 6 ]

Power 5 of 2 : $2^{5}$ = 32     [ Unit digit in 2 to power 5 is 2 ]

Power 6 of 2 : $2^{6}$ = 64     [ Unit digit in 2 to power 6 is 4 ]

Thus, the unit digits of values of powers of 2 follow the cycle of sequence:

2, 4, 8, 6, 2, 4, 8, 6, 2, 4...  and so on repeatedly.

Number of digits in the repeating cycle = 4 ( 2, 4, 8, 6 ). Thus, we get

1. Power of the form "4n+1" has unit digit = 2
2. Power of the form "4n+2" has unit digit = 4
3. Power of the form "4n+3" has unit digit = 8
4. Power of the form "4n" has unit digit = 6

We need to find the 727th power of 2. So, we represent 727 in the multiple of 4 forms:

           $727 = (4*181) + 3$

                  $= 4n + 3$     (where n = 181)

As the power of 2 is of the form "4n+3", the unit digit of the value will be 8.

Hence, the unit digit in the expansion of 2 to the power of 727 is 8.

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