Question

Ajutor va rog!
1) Determinati ultima cifra a numerelor:
a) $13^{74}$
b) $99^{51}$
c) $11^{53}$ + $15^{53}$ + $17^{53}$
d) $313^{100}$
e) $68^{86}$
f) $89^{37}$ + $88^{38}$ + $87^{39}$
g) $71^{10000001}$
h) $1009^{9001}$
i) $77^{36}$ + $77^{37}$ + $77^{38}$ + $77^{39}$
AJUTORR!

Answer 1

$a).13^7^4 \\ U(13^{74})=U(13^2 \cdot 13^7^2)=U(13^2 \cdot (13^4)^1^8)=U(169 \cdot 28561^1^8)= \\ =9 \cdot 1=9 \\ b).99^5^1 \\ U(99^5^1)=U(99 \cdot 99^5^0)=U(99\cdot (99^2)^2^5)=U(99 \cdot 9801^2^5)=9 \cdot 1=9 \\ c).11^5^3+15^5^3+17^5^3 \\ U(11^{53}+15^{53}+17^{53})=U(1+5+7)=U(13)=3 \\ d).313^{100} \\ U(313^{100})=U((313^4)^2^5)=U(3^4)^2^5)=U(81^2^5)=1 \\ e).68^8^6 \\ U(68^8^6)=U(68^2 \cdot 68^8^4)=U(68^2 \cdot (68^4)^2^1)=U(8^2 \cdot (8^4)^2^1)= \\ =U(64 \cdot 4096^2^1)=U(4 \cdot 6)=U(24)=4$
$f).89^{37}+88^{38}+87^{39} \\ U(89^{37}+88^{38}+87^{39})=U(9+4+3)=U(16)=6 \\ g).71^{10000001} \\ U(71^{10000001})=1 \\ h).1009^{9001} \\ U(1009^{9001})=U(1009 \cdot 1009^{9000})=U(1009 \cdot (1009^2)^{4500})= \\ =U(9 \cdot (9^2)^2^2^5^0)= U(9 \cdot 81^2^2^5^0)=9 \cdot 1=9 \\ i).77^3^6+77^3^7+77^3^8+77^3^9 \\ U(77^3^6+77^3^7+77^3^8+77^3^9)=U(1+7+9+3)=U(20)=0$