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Quantitative Aptitude Question on Divisibility and Remainder

When 3333 is divided by 11, the remainder is

A

1

B

6

C

5

D

10

Answer

5

Explanation

Solution

We need to find the remainder when 3333 is divided by 11.\text{We need to find the remainder when } 3^{333} \text{ is divided by } 11.

Using Fermat’s Little Theorem:\text{Using Fermat's Little Theorem:}
If p is a prime number and a is an integer such that a is not divisible by p, then ap−1≡1(modp).\text{If } p \text{ is a prime number and } a \text{ is an integer such that } a \text{ is not divisible by } p, \text{ then } a^{p-1} \equiv 1 \pmod{p}.

Here, p=11 and a=3. Fermat’s Little Theorem tells us that:\text{Here, } p = 11 \text{ and } a = 3. \text{ Fermat's Little Theorem tells us that:}
310≡1(mod11).3^{10} \equiv 1 \pmod{11}.

Now, we want to find 3333(mod11). First, reduce the exponent modulo 10:\text{Now, we want to find } 3^{333} \pmod{11}. \text{ First, reduce the exponent modulo } 10:
333÷10=33 remainder 3.333 \div 10 = 33 \text{ remainder } 3.

Thus, \text{Thus, }
3333≡33(mod11).3^{333} \equiv 3^3 \pmod{11}.

Next, calculate 33:\text{Next, calculate } 3^3:
33=27.3^3 = 27.

Now, find the remainder when 27 is divided by 11:\text{Now, find the remainder when 27 is divided by 11:}
27÷11=2 remainder 5.27 \div 11 = 2 \text{ remainder } 5.
Therefore, \text{Therefore, }
3333≡5(mod11).3^{333} \equiv 5 \pmod{11}.

5\boxed{5}