CAT 2024 Slot 1 QA Question 1

Multiple choice (+3 / −1) · Number Systems · Remainder · Try it, then check the answer and solution below.

When

10^{100}

is divided by 7, the remainder is

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Official Convenor Answer Key: C

Answer and solution

Answer: C) 4

📌 Core Concept

To solve the problem of finding the remainder when

10^{100}

is divided by 7, we use modular arithmetic and observe the repeating cycle of remainders.

🔢 Step-by-Step Solution

Find the remainders of smaller powers of 10 modulo 7:

10^1 \equiv 3 \mod 7

10^2 \equiv 2 \mod 7

10^3 \equiv 6 \mod 7

10^4 \equiv 4 \mod 7

10^5 \equiv 5 \mod 7

10^6 \equiv 1 \mod 7

Identify the cycle length:

The remainders repeat every 6 exponents:

3, 2, 6, 4, 5, 1

Determine the position of

10^{100}

in the cycle:

Divide 100 by 6:

100 = 6 \times 16 + 4

The remainder is 4, so

10^{100} \equiv 10^4 \mod 7

Find the remainder:

From the cycle,

10^4 \equiv 4 \mod 7

⚡ 30-Second Shortcut

Recognize that powers of 10 modulo 7 repeat every 6 exponents. Calculate \(100 \mod 6 =

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