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Question Number 176058 by Shrinava last updated on 11/Sep/22

$${22}^{22}$$$$\mathrm{what}\mathrm{is}\mathrm{hte}\mathrm{last}\mathrm{digit}?$$

Answered by BaliramKumar last updated on 11/Sep/22

$$4$$

Commented by BaliramKumar last updated on 11/Sep/22

$${22}^{22}={22}^{4\left(5\right)+2}[x^{4\left(n\right)+r}=x^r]0<r\le 4$$$$2^2=4Answer$$$$$$$$example{39857}^{2576}$$$${39857}^{4\left(n\right)+4}$$$$7^4=1Answer$$$$$$$$example{2345678}^{3566765!}$$$$8^{4\left(n\right)+4}=8^4=6$$$$$$$$Ifx!thenr=4(x\ge 4)$$

Commented by Shrinava last updated on 11/Sep/22

$$\mathrm{solution}\mathrm{please}$$

Commented by Rasheed.Sindhi last updated on 11/Sep/22

$$\mathrm{Your}\mathrm{rule}\mathrm{is}\mathbf{successful}!$$👍 $$\mathrm{Actually}\mathrm{I}\mathrm{think}:$$$$\mathrm{In}\mathrm{general}$$$$\mathrm{a}\equiv \mathrm{b}\left[10\right]\wedge \mathrm{m}\equiv \mathrm{n}\left[4\right]\iff {\mathrm{a}}^{\mathrm{m}}\equiv {\mathrm{b}}^{\mathrm{n}}\left[10\right]$$

Commented by Shrinava last updated on 13/Sep/22

$$\mathrm{cool}\mathrm{dear}\mathrm{sir}\mathrm{thankyou}$$

Answered by Rasheed.Sindhi last updated on 11/Sep/22

$$\mathrm{last}\mathrm{digit}={22}^{22}\mathrm{mod}10$$$$22\equiv 2\left(\mathrm{mod}10\right)$$$${22}^2\equiv 2^2\equiv 4\left(\mathrm{mod}10\right).............\left(\mathrm{i}\right)$$$${22}^5\equiv 2^5=32\equiv 2\left(\mathrm{mod}10\right)$$$${\left({22}^5\right)}^4\equiv 2^4=16\equiv 6\left(\mathrm{mod}10\right)$$$${22}^{20}\equiv 6\left(\mathrm{mod}10\right)..................\left(\mathrm{ii}\right)$$$$\left(\mathrm{i}\right)\times \left(\mathrm{ii}\right):$$$${22}^{22}\equiv 24\equiv 4\left(\mathrm{mod}10\right)$$

Answered by LordKazuma last updated on 11/Sep/22

$${22}^{22}\left(mod10\right)=2^{22}\left(mod10\right)$$$$={\left(2^3\right)}^7\cdot 2^1\left(mod10\right)$$$$={(-2)}^7\cdot 2\left(mod10\right)$$$$={\left(2^3\right)}^2\cdot (-2)\cdot 2\left(mod10\right)$$$$={(-2)}^2\cdot (-4)\left(mod10\right)$$$$=4\cdot (-4)\left(mod10\right)$$$$=(-16)\left(mod10\right)$$$$=(4-2\cdot 10)\left(mod10\right)=4$$$$sothelastdigitof{22}^{22}is4$$

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