CAT 2024 Slot 2 QA Question 2

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

When

3^{333}

is divide by 11, the remainder is

⌨️ Shortcuts:[1-4 / A-D] Select[Enter] Check[B] Star[S] Solution

Official Convenor Answer Key: D

Answer and solution

Answer: D) 5

📌 Core Concept

Fermat's Little Theorem states that if

p

is a prime number and

a

is an integer not divisible by

p , then:

a^{p-1} \equiv 1 \mod p $$

🔢 Step-by-Step Solution

Apply Fermat's Little Theorem:

Since 11 is prime and 3 is not divisible by 11:

3^{10} \equiv 1 \mod 11

Express 333 in terms of 10:

Divide 333 by 10:

333 = 10 \times 33 + 3

So,

3^{333} = 3^{10 \times 33 + 3} = (3^{10})^{33} \times 3^3

Simplify using Fermat's Theorem:

(3^{10})^{33} \equiv 1^{33} \equiv 1 \mod 11

Thus,

3^{333} \equiv 1 \times 3^3 \mod 11

Calculate

3^3

3^3 = 27

Find the remainder:

27 \div 11 = 2 \text{ with a remainder of } 5

So,

27 \equiv 5 \mod 11

⚡ 30-Second Shortcut

Use Fermat's Little Theorem to reduce the exponent modulo 10.

Compute

333 \mod 10 = 3

Calculate

3^3 = 27

Find

27 \mod 11 = 5

🎯 Final Answer

Correct Answer: Option D

Keep going