Question

Detetminati ultima cifra a nr :
99 la 51
11 la 53+ 15 la 53+ 17 la 53
313 la 100
68 la 86
89 la 37+88 la 38+87 la 39
71 la 10000001
1009 la 9001
77 la 36+ 77 la 37+77 la 38+77 la 39
Va rog mult ajutati ma aveti coroana si 20 de puncte

Answer 1

$a).99^5^1 \\ U(99^5^1)=U(99 \times 99^5^0)=U(99 \times (99 ^2)^2^5)=U(99 \times 9801^2^5)= \\ =9 \times 1=9 \\ \\ b).11^5^3+15^5^3+17^5^3 \\ U(11^5^3+15^5^3+17^5^3)=U(1+5+7)=U(13)=3 \\ \\ c).313^1^0^0 \\ U(313^{100})=U((313^4)^2^5)=U((3^4)^2^5)=U(81^2^5)=1 \\ \\ d).68^8^6 \\ U(68^8^6)=U(68^2 \times 68^8^4)=U(68^2 \times (68^4)^2^1)=U(8^2 \times (8^4)^2^1)= \\ =U(64 \times 4096^2^1)=U(4 \times 6)=U(24)=4$

$e). 89^3^7+88^3^8+87^3^9 \\ U(89^3^7+88^3^8+87^3^9)=U(9+4+3)=U(16)=6 \\ \\ f).71^{10000001} \\ U(71^{10000001})=1 \\ \\ g).1009^{9001} \\ U(1009^{9001})=U(1009 \times 1009^{9000})=U(1009 \times (1009^2)^{4500})= \\ =U(9 \times (9^2)^{2250})=U(9 \times 81^{4500})=9 \times 1=9 \\ \\ h).77^3^6+77^3^7+77^3^8+77^3^9 \\ U(77^3^6+77^3^7+77^3^8+77^3^9)=U(1+7+9+3)=U(20)=0$