Question

If 3^2017 is divided by 10, then the remainder is

Answer 1

Answer:

3 will be the answer

...

.because 3^ any odd number leaves the answer 3

Answer 2

Remainder of division is 3, when $\red{ {3}^{2017}}$ is divided by 10.

Step-by-step explanation:

Given:

• ${3}^{2017}$

To find:

• Find the remainder when given number is divided by 10.

Solution:

Concept to be used:

• Unit digit will be the remainder, when a number is divided by 10.
• To find unit digit of given number, find the periodicity of power of 3.

Step 1:

Find the periodicity of powers of 3.

We know that,

${3}^{1} =\underline3$

${3}^{2} =\underline9$

${3}^{3} = 2\underline 7$

${3}^{4} = 8 \underline1$

${3}^{5} = 24 \underline3$

it is visible that, after 4 powers of 3, the numbers of unit place will be repeating.

i.e. Unit place digits are 3,9,7,1,3,9,7,1,...

Thus,

Periodicity is 4.

Step 2:

Divide the power by 4.

$4)2017(504 \\ 20 \\ - - - - \\ 17 \\ 16 \\ - - - - \\ 1 \\ - - - -$

Thus,

${3}^{2017} = {3}^{(4 \times 504)} \times {3}^{1}$

The unit place digit of ${3}^{4 \times 504}$ will be 1.

*Remark : ${3}^{4 \times 504}=({3}^4)^{504}={81}^{504}$, one can understand like this 81 have unit place digit '1', any power to this will give '1' at unit place digit.

When it is multiplied with 3¹, the unit place digit of the number ${3}^{2017}$ is 3.

Thus,

Remainder of division is 3, when $\bf\:{3}^{2017}$is divided by 10.

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