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2018-2019-2019-2018-mod-4-

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Question Number 89586 by Asif Hypothesis last updated on 18/Apr/20
2018^(2019) −2019^(2018 ) ≡? (mod 4)
20182019−20192018≡?(mod4)
Answered by Asif Hypothesis last updated on 18/Apr/20
We observe, 2018 ≡ 2 (mod 4)⇒2018^(2019) ≡ 2^(2019) ≡ 0 (mod 4) (∗) 2019 ≡ 3 (mod 4)⇒2019^(2018) ≡ 3^(2018) ≡(3^2 )^(1009) ≡ (1)^(1009) ≡1(mod 4) (∗∗) (∗)+(∗∗) yields 2018^(2019) −2019^(2018) ≡ 0+1 ≡1 (mod 4)
Weobserve,2018≡2(mod4)⇒20182019≡22019≡0(mod4)(∗)2019≡3(mod4)⇒20192018≡32018≡(32)1009≡(1)1009≡1(mod4)(∗∗)(∗)+(∗∗)yields20182019−20192018≡0+1≡1(mod4)

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