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UPSC CDS II 2024 Math Paper

Question : 73 of 100

Marks: +1, -0

What is the remainder if we divide 3^{10} by 7?

Solution:

Concept:

Modular arithmetic helps find remainders by breaking large exponents into smaller steps using patterns in powers modulo the divisor.

Explanation:

We need

3^{10} \mod 7

Compute powers of

3

modulo

7

step by step:

3^1 \mod 7 = 3

3^2 \mod 7 = (3 \times 3) = 9 \equiv 2

3^3 \mod 7 = (3 \times 2) = 6 \equiv 6

3^4 \mod 7 = (3 \times 6) = 18 \equiv 4

3^5 \mod 7 = (3 \times 4) = 12 \equiv 5

3^6 \mod 7 = (3 \times 5) = 15 \equiv 1

Now

3^6 \equiv 1 \pmod{7}

, so higher powers repeat every

6

cycles.

Write

3^{10} = 3^6 \times 3^4

Then

3^{10} \equiv 1 \times 4 = 4 \pmod{7}

Thus, the remainder is

4

Answer:

4

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