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a-12-223-7-56-19-25-what-is-the-last-digit-of-the-number-

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Question Number 224146 by hardmath last updated on 22/Aug/25
a = 12^(223) ∙ 7^(56) + 19^(25) what is the last digit of the number?
$$\boldsymbol{\mathrm{a}}\:=\:\mathrm{12}^{\mathrm{223}} \:\centerdot\:\mathrm{7}^{\mathrm{56}} \:+\:\mathrm{19}^{\mathrm{25}} \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{last}\:\mathrm{digit}\:\mathrm{of}\:\mathrm{the}\:\mathrm{number}? \\ $$
Answered by Ghisom_ last updated on 23/Aug/25
12^(223) has the same last digit as 2^(223) 2^(4n+3) has the last digit 8 7^(4n) has the last digit 1 ⇒ 12^(223) ×7^(56) has the last digit 8 19^(25) has the same last digit as 9^(25) 9^(2n+1) has the last digit 9 8+9=17 7
$$\mathrm{12}^{\mathrm{223}} \:\mathrm{has}\:\mathrm{the}\:\mathrm{same}\:\mathrm{last}\:\mathrm{digit}\:\mathrm{as}\:\mathrm{2}^{\mathrm{223}} \\ $$$$\mathrm{2}^{\mathrm{4}{n}+\mathrm{3}} \:\mathrm{has}\:\mathrm{the}\:\mathrm{last}\:\mathrm{digit}\:\mathrm{8} \\ $$$$\mathrm{7}^{\mathrm{4}{n}} \:\mathrm{has}\:\mathrm{the}\:\mathrm{last}\:\mathrm{digit}\:\mathrm{1} \\ $$$$\Rightarrow\:\mathrm{12}^{\mathrm{223}} ×\mathrm{7}^{\mathrm{56}} \:\mathrm{has}\:\mathrm{the}\:\mathrm{last}\:\mathrm{digit}\:\mathrm{8} \\ $$$$\mathrm{19}^{\mathrm{25}} \:\mathrm{has}\:\mathrm{the}\:\mathrm{same}\:\mathrm{last}\:\mathrm{digit}\:\mathrm{as}\:\mathrm{9}^{\mathrm{25}} \\ $$$$\mathrm{9}^{\mathrm{2}{n}+\mathrm{1}} \:\mathrm{has}\:\mathrm{the}\:\mathrm{last}\:\mathrm{digit}\:\mathrm{9} \\ $$$$\mathrm{8}+\mathrm{9}=\mathrm{17} \\ $$$$\mathrm{7} \\ $$
Commented by hardmath last updated on 23/Aug/25
thank you dear professor... 4n + 3 4n 2n + 1 where did what I mentioned come from.?
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{dear}\:\mathrm{professor}… \\ $$$$\mathrm{4}\boldsymbol{\mathrm{n}}\:+\:\mathrm{3} \\ $$$$\mathrm{4}\boldsymbol{\mathrm{n}} \\ $$$$\mathrm{2}\boldsymbol{\mathrm{n}}\:+\:\mathrm{1} \\ $$$$\mathrm{where}\:\mathrm{did}\:\mathrm{what}\:\mathrm{I}\:\mathrm{mentioned}\:\mathrm{come}\:\mathrm{from}.? \\ $$
Commented by Ghisom_ last updated on 23/Aug/25
2^1 =2; 2^2 =4; 2^3 =8; 2^4 =...6; 2^5 =...2 ⇒ 2^(4n) =...6; 2^(4n+1) =...2; 2^(4n+2) =...4; 2^(4n+3) =...8 7^1 =7; 7^2 =...9; 7^3 =...3; 7^4 =...1; 7^5 =...7 ⇒ 7^(4n) =...1; 7^(4n+1) =...7; 7^(4n+2) =...9; 7^(4n+3) =...3 9^1 =9; 9^2 =...1; 9^3 =...9 ⇒ 9^(2n) =...1; 9^(2n+1) =...9
$$\mathrm{2}^{\mathrm{1}} =\mathrm{2};\:\mathrm{2}^{\mathrm{2}} =\mathrm{4};\:\mathrm{2}^{\mathrm{3}} =\mathrm{8};\:\mathrm{2}^{\mathrm{4}} =…\mathrm{6};\:\mathrm{2}^{\mathrm{5}} =…\mathrm{2} \\ $$$$\Rightarrow \\ $$$$\mathrm{2}^{\mathrm{4}{n}} =…\mathrm{6};\:\mathrm{2}^{\mathrm{4}{n}+\mathrm{1}} =…\mathrm{2};\:\mathrm{2}^{\mathrm{4}{n}+\mathrm{2}} =…\mathrm{4};\:\mathrm{2}^{\mathrm{4}{n}+\mathrm{3}} =…\mathrm{8} \\ $$$$ \\ $$$$\mathrm{7}^{\mathrm{1}} =\mathrm{7};\:\mathrm{7}^{\mathrm{2}} =…\mathrm{9};\:\mathrm{7}^{\mathrm{3}} =…\mathrm{3};\:\mathrm{7}^{\mathrm{4}} =…\mathrm{1};\:\mathrm{7}^{\mathrm{5}} =…\mathrm{7} \\ $$$$\Rightarrow \\ $$$$\mathrm{7}^{\mathrm{4}{n}} =…\mathrm{1};\:\mathrm{7}^{\mathrm{4}{n}+\mathrm{1}} =…\mathrm{7};\:\mathrm{7}^{\mathrm{4}{n}+\mathrm{2}} =…\mathrm{9};\:\mathrm{7}^{\mathrm{4}{n}+\mathrm{3}} =…\mathrm{3} \\ $$$$ \\ $$$$\mathrm{9}^{\mathrm{1}} =\mathrm{9};\:\mathrm{9}^{\mathrm{2}} =…\mathrm{1};\:\mathrm{9}^{\mathrm{3}} =…\mathrm{9} \\ $$$$\Rightarrow \\ $$$$\mathrm{9}^{\mathrm{2}{n}} =…\mathrm{1};\:\mathrm{9}^{\mathrm{2}{n}+\mathrm{1}} =…\mathrm{9} \\ $$

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