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Question Number 206025 by cortano12 last updated on 05/Apr/24

$$2^{2024}=x\left(mod10\right)$$

Answered by BaliramKumar last updated on 05/Apr/24

$$\mathrm{Find}\mathrm{unit}\mathrm{digit}$$$$2^{2024}=2^{4\mathrm{k}+4}=2^4=16\Rightarrow 6$$

Answered by Rasheed.Sindhi last updated on 05/Apr/24

$$2^{2024}=x\left(mod10\right)$$$$2^4\equiv 6\left(mod10\right)$$$$2^{2024}\equiv x\left(mod10\right)$$$$\Rightarrow 2^{4\times 506}\equiv x\left(mod10\right)$$$$\Rightarrow {\left(2^4\right)}^{506}\equiv x\left(mod10\right)$$$$\Rightarrow {\left(6\right)}^{506}\equiv x\left(mod10\right)$$$$Observethat{6}^k\equiv 6\left(mod10\right)\mathrm{\forall }k\in \mathbb{N}$$$$\therefore 6^{506}\equiv 6\left(mod10\right)$$$$\therefore {\left(2^4\right)}^{506}\equiv 6\left(mod10\right)$$$$\therefore 2^{2024}\equiv 6\left(mod10\right)$$

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