2-2024-x-mod-10-

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

$$\:\:\:\:\:\mathrm{2}^{\mathrm{2024}} \:=\:{x}\:\left({mod}\:\mathrm{10}\right)\: \\ $$

Answered by BaliramKumar last updated on 05/Apr/24

Find unit digit 2^(2024) = 2^(4k + 4) = 2^4 = 16 ⇒ 6

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

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

2^(2024) = x (mod 10) 2^4 ≡6(mod 10) 2^(2024) ≡ x (mod 10) ⇒2^(4×506) ≡x(mod 10) ⇒(2^4 )^(506) ≡x(mod 10) ⇒(6)^(506) ≡x(mod 10) Observe that 6^k ≡6(mod 10) ∀k∈N ∴ 6^(506) ≡6 (mod 10) ∴ (2^4 )^(506) ≡6 (mod 10) ∴ 2^(2024) ≡6(mod 10)

$$\:\:\:\:\:\mathrm{2}^{\mathrm{2024}} \:=\:{x}\:\left({mod}\:\mathrm{10}\right)\: \\ $$$$\:\:\:\mathrm{2}^{\mathrm{4}} \equiv\mathrm{6}\left({mod}\:\mathrm{10}\right) \\ $$$$\:\mathrm{2}^{\mathrm{2024}} \:\equiv\:{x}\:\left({mod}\:\mathrm{10}\right) \\ $$$$\Rightarrow\mathrm{2}^{\mathrm{4}×\mathrm{506}} \equiv{x}\left({mod}\:\mathrm{10}\right) \\ $$$$\Rightarrow\left(\mathrm{2}^{\mathrm{4}} \right)^{\mathrm{506}} \equiv{x}\left({mod}\:\mathrm{10}\right) \\ $$$$\Rightarrow\left(\mathrm{6}\right)^{\mathrm{506}} \equiv{x}\left({mod}\:\mathrm{10}\right) \\ $$$${Observe}\:{that}\:\mathrm{6}^{{k}} \equiv\mathrm{6}\left({mod}\:\mathrm{10}\right)\:\forall{k}\in\mathbb{N} \\ $$$$\therefore\:\mathrm{6}^{\mathrm{506}} \equiv\mathrm{6}\:\left({mod}\:\mathrm{10}\right) \\ $$$$\therefore\:\left(\mathrm{2}^{\mathrm{4}} \right)^{\mathrm{506}} \equiv\mathrm{6}\:\left({mod}\:\mathrm{10}\right) \\ $$$$\therefore\:\mathrm{2}^{\mathrm{2024}} \equiv\mathrm{6}\left({mod}\:\mathrm{10}\right) \\ $$

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