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Question Number 176058 by Shrinava last updated on 11/Sep/22
22^(22) what is hte last digit?
$$\mathrm{22}^{\mathrm{22}} \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{hte}\:\mathrm{last}\:\mathrm{digit}? \\ $$
Answered by BaliramKumar last updated on 11/Sep/22
4
$$\mathrm{4} \\ $$
Commented by BaliramKumar last updated on 11/Sep/22
22^(22) = 22^(4(5)+2) [x^(4(n) + r) = x^r ] 0<r≤4 2^2 = 4 Answer example 39857^(2576) 39857^(4(n)+4) 7^4 = 1 Answer example 2345678^(3566765!) 8^(4(n)+4) = 8^4 = 6 If x! then r=4 (x ≥ 4)
$$\mathrm{22}^{\mathrm{22}} \:=\:\mathrm{22}^{\mathrm{4}\left(\mathrm{5}\right)+\mathrm{2}} \:\:\:\:\:\:\:\left[{x}^{\mathrm{4}\left({n}\right)\:+\:{r}} \:=\:{x}^{{r}} \right]\:\:\mathrm{0}<{r}\leq\mathrm{4} \\ $$$$\mathrm{2}^{\mathrm{2}} \:=\:\mathrm{4}\:{Answer} \\ $$$$ \\ $$$${example}\:\mathrm{39857}^{\mathrm{2576}} \\ $$$$\mathrm{39857}^{\mathrm{4}\left({n}\right)+\mathrm{4}} \\ $$$$\mathrm{7}^{\mathrm{4}} \:=\:\mathrm{1}\:{Answer} \\ $$$$ \\ $$$${example}\:\:\:\:\:\:\:\mathrm{2345678}^{\mathrm{3566765}!} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{8}^{\mathrm{4}\left({n}\right)+\mathrm{4}} \:=\:\mathrm{8}^{\mathrm{4}} \:=\:\mathrm{6} \\ $$$$ \\ $$$${If}\:\:\:\:{x}!\:\:\:\:{then}\:\:\:\:{r}=\mathrm{4}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({x}\:\geq\:\mathrm{4}\right) \\ $$
Commented by Shrinava last updated on 11/Sep/22
solution please
$$\mathrm{solution}\:\mathrm{please} \\ $$
Commented by Rasheed.Sindhi last updated on 11/Sep/22
Your rule is successful! 👍 Actually I think: In general a≡b[10] ∧ m≡n[4]⇔a^m ≡b^n [10]
$$\mathrm{Your}\:\mathrm{rule}\:\mathrm{is}\:\boldsymbol{\mathrm{successful}}!\: \\ $$👍
$$\mathrm{Actually}\:\mathrm{I}\:\mathrm{think}: \\ $$$$\mathrm{In}\:\mathrm{general} \\ $$$$\mathrm{a}\equiv\mathrm{b}\left[\mathrm{10}\right]\:\wedge\:\mathrm{m}\equiv\mathrm{n}\left[\mathrm{4}\right]\Leftrightarrow\mathrm{a}^{\mathrm{m}} \equiv\mathrm{b}^{\mathrm{n}} \left[\mathrm{10}\right] \\ $$
Commented by Shrinava last updated on 13/Sep/22
cool dear sir thankyou
$$\mathrm{cool}\:\mathrm{dear}\:\mathrm{sir}\:\mathrm{thankyou} \\ $$
Answered by Rasheed.Sindhi last updated on 11/Sep/22
last digit=22^(22) mod 10 22≡2(mod 10) 22^2 ≡2^2 ≡4(mod 10).............(i) 22^5 ≡2^5 =32≡2(mod 10) (22^5 )^4 ≡2^4 =16≡6(mod 10) 22^(20) ≡6(mod 10)..................(ii) (i)×(ii): 22^(22) ≡24≡4(mod 10)
$$\mathrm{last}\:\mathrm{digit}=\mathrm{22}^{\mathrm{22}} \mathrm{mod}\:\mathrm{10} \\ $$$$\mathrm{22}\equiv\mathrm{2}\left(\mathrm{mod}\:\mathrm{10}\right) \\ $$$$\mathrm{22}^{\mathrm{2}} \equiv\mathrm{2}^{\mathrm{2}} \equiv\mathrm{4}\left(\mathrm{mod}\:\mathrm{10}\right)………….\left(\mathrm{i}\right) \\ $$$$\mathrm{22}^{\mathrm{5}} \equiv\mathrm{2}^{\mathrm{5}} =\mathrm{32}\equiv\mathrm{2}\left(\mathrm{mod}\:\mathrm{10}\right) \\ $$$$\left(\mathrm{22}^{\mathrm{5}} \right)^{\mathrm{4}} \equiv\mathrm{2}^{\mathrm{4}} =\mathrm{16}\equiv\mathrm{6}\left(\mathrm{mod}\:\mathrm{10}\right) \\ $$$$\mathrm{22}^{\mathrm{20}} \equiv\mathrm{6}\left(\mathrm{mod}\:\mathrm{10}\right)………………\left(\mathrm{ii}\right) \\ $$$$\left(\mathrm{i}\right)×\left(\mathrm{ii}\right): \\ $$$$\mathrm{22}^{\mathrm{22}} \equiv\mathrm{24}\equiv\mathrm{4}\left(\mathrm{mod}\:\mathrm{10}\right) \\ $$
Answered by LordKazuma last updated on 11/Sep/22
22^(22) (mod 10) = 2^(22) (mod 10) = (2^3 )^7 ∙ 2^1 (mod 10) = (−2)^7 ∙ 2 (mod 10) = (2^3 )^2 ∙ (−2) ∙ 2 (mod 10) = (−2)^2 ∙ (−4) (mod 10) = 4 ∙ (−4) (mod 10) = (−16) (mod 10) = (4 − 2 ∙ 10) (mod 10) = 4 so the last digit of 22^(22) is 4
$$\mathrm{22}^{\mathrm{22}} \:\left({mod}\:\mathrm{10}\right)\:=\:\mathrm{2}^{\mathrm{22}} \:\left({mod}\:\mathrm{10}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\left(\mathrm{2}^{\mathrm{3}} \right)^{\mathrm{7}} \:\centerdot\:\mathrm{2}^{\mathrm{1}} \:\left({mod}\:\mathrm{10}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\left(−\mathrm{2}\right)^{\mathrm{7}} \:\centerdot\:\mathrm{2}\:\left({mod}\:\mathrm{10}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\left(\mathrm{2}^{\mathrm{3}} \right)^{\mathrm{2}} \:\centerdot\:\left(−\mathrm{2}\right)\:\centerdot\:\mathrm{2}\:\left({mod}\:\mathrm{10}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\left(−\mathrm{2}\right)^{\mathrm{2}} \:\centerdot\:\left(−\mathrm{4}\right)\:\left({mod}\:\mathrm{10}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{4}\:\centerdot\:\left(−\mathrm{4}\right)\:\left({mod}\:\mathrm{10}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\left(−\mathrm{16}\right)\:\left({mod}\:\mathrm{10}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\left(\mathrm{4}\:−\:\mathrm{2}\:\centerdot\:\mathrm{10}\right)\:\left({mod}\:\mathrm{10}\right)\:=\:\mathrm{4} \\ $$$${so}\:{the}\:{last}\:{digit}\:{of}\:\mathrm{22}^{\mathrm{22}} \:{is}\:\mathrm{4} \\ $$

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