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UPSC CDS 1 2021 Math Paper

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Question : 96 of 100

Marks: +1, -0

What is the remainder when

27^{27} - 15^{27}

is divided by 6?

0
1
3
4

Solution:

Concept:

For odd exponent

n

,

a^n - b^n

is divisible by

(a - b)

. If

(a - b)

is itself divisible by

6

, then the expression is divisible by

6

.

Explanation:

Here,

27^{27} - 15^{27}

is in the form

a^n - b^n

with

n = 27

(odd).

a = 27

,

b = 15

, so

a - b = 12

.

Since

12

is divisible by

6

, the expression

27^{27} - 15^{27}

is divisible by

6

.

Alternatively, note that

27 \equiv 3 \pmod{6}

and

15 \equiv 3 \pmod{6}

.

Thus,

27^{27} \equiv 3^{27} \pmod{6}

and

15^{27} \equiv 3^{27} \pmod{6}

.

So

27^{27} - 15^{27} \equiv 3^{27} - 3^{27} \equiv 0 \pmod{6}

.

Therefore, the remainder is

0

.

Answer:

0

(Option A)

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