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Question Number 89586 by Asif Hypothesis last updated on 18/Apr/20

$$$$$${2018}^{2019}-{2019}^{2018}\equiv ?\left(mod4\right)$$

Answered by Asif Hypothesis last updated on 18/Apr/20

$$Weobserve,$$$$2018\equiv 2\left(mod4\right)\Rightarrow {2018}^{2019}\equiv 2^{2019}\equiv 0\left(mod4\right)(\ast )$$$$2019\equiv 3\left(mod4\right)\Rightarrow {2019}^{2018}\equiv 3^{2018}\equiv {\left(3^2\right)}^{1009}\equiv {\left(1\right)}^{1009}\equiv 1\left(mod4\right)(\ast \ast )$$$$(\ast )+(\ast \ast )yields$$$${2018}^{2019}-{2019}^{2018}\equiv 0+1\equiv 1\left(mod4\right)$$$$$$$$$$

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