Question-189700

Question Number 189700 by DAVONG last updated on 20/Mar/23

Commented by Rasheed.Sindhi last updated on 21/Mar/23

$${a}=\mathrm{21} \\ $$

Commented by mehdee42 last updated on 21/Mar/23

it^, s wrong .because 21^5 ≡^(100) 01⇒21^(2020) ≡^(100) 01⇒21^(2022) ≡^(100) 21^2 ≡^(100) 41

$${it}^{,} {s}\:{wrong}\:.{because} \\ $$$$\mathrm{21}^{\mathrm{5}} \overset{\mathrm{100}} {\equiv}\mathrm{01}\Rightarrow\mathrm{21}^{\mathrm{2020}} \overset{\mathrm{100}} {\equiv}\mathrm{01}\Rightarrow\mathrm{21}^{\mathrm{2022}} \overset{\mathrm{100}} {\equiv}\mathrm{21}^{\mathrm{2}} \overset{\mathrm{100}} {\equiv}\mathrm{41} \\ $$

Answered by mehdee42 last updated on 21/Mar/23

11^2 ≡^(100) 21⇒11^(10) ≡^(100) 21^5 ≡^(100) 01 ⇒11^(2020) ≡^(100) 01⇒11^(2022) ≡^(100) 21⇒a=11

$$\mathrm{11}^{\mathrm{2}} \overset{\mathrm{100}} {\equiv}\mathrm{21}\Rightarrow\mathrm{11}^{\mathrm{10}} \overset{\mathrm{100}} {\equiv}\mathrm{21}^{\mathrm{5}} \overset{\mathrm{100}} {\equiv}\mathrm{01} \\ $$$$\Rightarrow\mathrm{11}^{\mathrm{2020}} \overset{\mathrm{100}} {\equiv}\mathrm{01}\Rightarrow\mathrm{11}^{\mathrm{2022}} \overset{\mathrm{100}} {\equiv}\mathrm{21}\Rightarrow{a}=\mathrm{11} \\ $$

Commented by Rasheed.Sindhi last updated on 22/Mar/23

$${It}'{s}\:{a}^{\mathrm{2021}} ,\:{not}\:{a}^{\mathrm{2022}} \\ $$

Commented by Rasheed.Sindhi last updated on 22/Mar/23

21^5 ≡1(mod 100) 21^(2020) ≡1(mod 100) 21^(2020) ×21≡21(mod 100) 21^(2021) ≡21(mod 100)

$$\mathrm{21}^{\mathrm{5}} \equiv\mathrm{1}\left({mod}\:\mathrm{100}\right) \\ $$$$\mathrm{21}^{\mathrm{2020}} \equiv\mathrm{1}\left({mod}\:\mathrm{100}\right) \\ $$$$\mathrm{21}^{\mathrm{2020}} ×\mathrm{21}\equiv\mathrm{21}\left({mod}\:\mathrm{100}\right) \\ $$$$\mathrm{21}^{\mathrm{2021}} \equiv\mathrm{21}\left({mod}\:\mathrm{100}\right) \\ $$

Answered by talminator2856792 last updated on 22/Mar/23

... X Y 1 × 11 = ... X Y 1 0 + ... Y 1 ^� ^� ^� ^� Y^� +1 ^� 1^� ⇒ 11^(...xyz1) = ...z1 11^(2021) = ...21 a = 11

$$\:…\:\mathrm{X}\:\mathrm{Y}\:\mathrm{1}\:×\:\mathrm{11}\:=\:\:…\:\mathrm{X}\:\mathrm{Y}\:\mathrm{1}\:\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\:\:\:…\:\mathrm{Y}\:\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\bar {\:}\bar {\:}\bar {\:}\bar {\:}\bar {\mathrm{Y}}+\mathrm{1}\bar {\:}\bar {\mathrm{1}} \\ $$$$\: \\ $$$$\:\:\Rightarrow\:\:\mathrm{11}^{…\mathrm{xyz1}} \:=\:…\mathrm{z1} \\ $$$$\: \\ $$$$\:\:\mathrm{11}^{\mathrm{2021}} \:=\:…\mathrm{21} \\ $$$$\:\:{a}\:=\:\mathrm{11} \\ $$$$\: \\ $$

Commented by Rasheed.Sindhi last updated on 22/Mar/23

11^(2021) ≠...21 11^(2021) =...11 : 11^(10) ≡1(mod 100) (11^(10) )^(202) ≡1^(202) (mod 100) (11^(2020) ×11≡1×11(mod 100) 11^(2021) ≡11(mod 11)

$$\mathrm{11}^{\mathrm{2021}} \neq…\mathrm{21} \\ $$$$\mathrm{11}^{\mathrm{2021}} =…\mathrm{11}\:: \\ $$$$\mathrm{11}^{\mathrm{10}} \equiv\mathrm{1}\left({mod}\:\mathrm{100}\right) \\ $$$$\left(\mathrm{11}^{\mathrm{10}} \right)^{\mathrm{202}} \equiv\mathrm{1}^{\mathrm{202}} \left({mod}\:\mathrm{100}\right) \\ $$$$\left(\mathrm{11}^{\mathrm{2020}} ×\mathrm{11}\equiv\mathrm{1}×\mathrm{11}\left({mod}\:\mathrm{100}\right)\right. \\ $$$$\mathrm{11}^{\mathrm{2021}} \equiv\mathrm{11}\left({mod}\:\mathrm{11}\right) \\ $$$$ \\ $$

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