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JEE Main 202529 JanuaryEvening Shift

Exam: JEE Main
Year: 2025
Date: 29 January
Shift: Evening Shift
Subject: Mathematics
Chapter: Binomial Theorem

JEE Main 2025 · 29 January · Evening Shift · Mathematics · Binomial Theorem

The remainder, when 7^ 103 is divided by 23 , is equal to :

The remainder, when

7^{103}

is divided by 23 , is equal to :

Options

A6
B17
C9
D14

Correct answer

D. 14

Step-by-step solution

\begin{aligned} & 7^{103}=7\left(7^{102}\right)=7(343)^{34}=7(345-2)^{34} \\ & 7^{103}=23 \mathrm{ K}_1+7.2^{34} \end{aligned}

Now

7.2^{34}=7.2^2 \cdot 2^{32}

\begin{aligned} & =28 \cdot(256)^4 \\ & =28(253+3)^4 \\ & \therefore 28 \times 81 \Rightarrow(23+5)(69+12) \\ & 23 \mathrm{ K}_2+60 \\ & \therefore \text{ Remainder }=14 \end{aligned}

$$

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