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UPSC CDS 2 2020 Math Paper

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Question : 8 of 100

Marks: +1, -0

What is the digit in the unit place of

3^{99}

?

1
3
7
9

Solution:

Concept:

The unit digit of powers of 3 repeats in a cycle of 4: 3, 9, 7, 1.

Explanation:

The unit digits of

3^{n}

for

n = 1,2,3,4

are

3, 9, 7, 1

respectively.

After every 4 powers, the pattern repeats.

We write

3^{99}

as

3^{96} \times 3^{3}

because

96

is a multiple of

4

.

3^{96} = (3^{4})^{24}

. The unit digit of

3^{4}

is

1

, so

(1)^{24} = 1

.

3^{3} = 27

, so its unit digit is

7

.

Thus the unit digit of

3^{99}

is

1 \times 7 = 7

.

Answer:

C. 7

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