Model Paper 6 — Applied Maths STD 12 Science — Question
CBSE BoardEnglish MediumSTD 12 ScienceApplied MathsModel Paper 62 Marks
Question
Prove that: 3500 ≡ 2 (mod 7)
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Answer
In order to prove $3^{500} \equiv 2(\bmod 7)$, let us first find an integer $k$ such that $3^k \equiv \pm 1(\bmod 7)$. We know that $3^1 \equiv 3(\bmod 7)$ $ \begin{array}{l} \Rightarrow 3^2 \equiv 3 \times 3=9 \equiv 2(\bmod 7) \\ \Rightarrow 3^3 \equiv 3 \times 2(\bmod 7) \\ \Rightarrow 3^3 \equiv 6 \equiv-1(\bmod 7) \end{array} $ Thus, we find that $3^3 \equiv-1(\bmod 7)$. Let us now express $3^{500}$ in terms of $3^3$. $ 3^{500}=\left(3^3\right)^{166} \times 3^2 $ Now, $3^3 \equiv-1(\bmod 7)$ $\Rightarrow\left(3^3\right)^{166} \equiv(-1)^{166}(\bmod 7)\left[\because a \equiv b(\bmod m) \Rightarrow a^n \equiv b^n(\bmod m)\right]$ $\Rightarrow\left(3^3\right)^{166} \times 3^2 \equiv(-1)^{166} \times 3^2(\bmod 7)[\because a \equiv b(\bmod m) \Rightarrow a x \equiv b x(\bmod m)]$ $\Rightarrow 3^{500} \equiv 9(\bmod 7)$ But, $9 \equiv 2(\bmod 7)$. Thus, we obtain $ \begin{array}{l} 3^{500} \equiv 9(\bmod 7) \text { and } 9 \equiv 2(\bmod 7) \\ \Rightarrow 3^{500} \equiv 2(\bmod 7)[\because a \equiv b(\bmod m), b \equiv c(\bmod m) \Rightarrow a \equiv c(\bmod m)] \end{array} $
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