Question

Determinati ultima cifra a numerelor:
a) $99^{51}$
b) $11^{53} + 15^{53} + 17^{53}$
c) $313^{100}$
d) $68^{86}$
f) $89^{37} + 88^{38} + 87^{39}$
g) $71^{10000001}$
h) $1009^{9001}$
i) $77^{36} + 77^{37} + 77^{38} + 77^{39}$

Answer 1

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

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