Fractional part of the number 4^202215 is equal to: — Solution | JEE Mathematics

MCQEasyJEE 2023Complex Numbers Basics

Fractional part of the number $\frac{4^{2022}}{15}$ is equal to:

Answer

Correct answer:C

Standard Method

Given: We need the fractional part of $\frac{4^{2022}}{15}$ .

Find: The fractional part of the given number.

Use the remainder when $4^{2022}$ is divided by $15$ .

4^1 \bmod 15 = 4

4^2 \bmod 15 = 16 \bmod 15 = 1

So the powers of $4$ modulo $15$ repeat with period $2$ :

4^1 \equiv 4 \pmod{15}, \quad 4^2 \equiv 1 \pmod{15}, \quad 4^3 \equiv 4 \pmod{15}, \quad 4^4 \equiv 1 \pmod{15}

Since $2022$ is even,

4^{2022} \equiv 1 \pmod{15}

Therefore,

\frac{4^{2022}}{15} = \text{integer part} + \frac{1}{15}

Hence, the fractional part is $\frac{1}{15}$ . The correct option is C.

Remainder Trick

Given: The number is $\frac{4^{2022}}{15}$ .

Find: Its fractional part.

For any expression of the form $\frac{N}{15}$ , if the remainder when $N$ is divided by $15$ is $r$ , then the fractional part is $\frac{r}{15}$ .

Here,

4^2 = 16 \equiv 1 \pmod{15}

So,

4^{2022} = \left(4^2\right)^{1011} \equiv 1^{1011} = 1 \pmod{15}

Thus the remainder is $1$ , and the fractional part is $\frac{1}{15}$ . Therefore, the correct option is C.

Assuming the fractional part depends on the full value of $4^{2022}$ instead of only its remainder modulo $15$ . This is wrong because for $\frac{N}{15}$ , the fractional part is determined by the remainder on division by $15$ . First find $N \bmod 15$ .
Missing the periodic pattern in powers of $4$ modulo $15$ . This leads to unnecessary computation. Since $4^2 \equiv 1 \pmod{15}$ , use parity of the exponent instead of expanding large powers.
Confusing the remainder $1$ with the fractional part itself. The remainder is not the answer directly; the fractional part is $\frac{\text{remainder}}{15} = \frac{1}{15}$ .

Get unlimited AI-adaptive practice, mastery tracking, and an AI tutor that explains every step — free to start.