Question

a) u(89³⁷+88³⁸+37³⁹)=
b) u(71¹⁰⁰⁰⁰⁰⁰¹)=
c) u(1009⁹⁰⁰¹)=
d) u(77³⁶+77³⁷+77³⁸+77³⁹)=
*u=ultima cifra

Answer 1

a)
$U( 89^{37} + 88^{38} + 37^{39}) =$

$= U( 89^{2 * 18 + 1} + 88^{4 * 9 + 2} + 37^{4*9+3}) =$

$= U( 7921^{18}*89 + 59969536^{9} * 7744 + 1874161^{9} * 50653 ) =$

$= U(1 * 9 + 6*4 + 1 * 3) = U(9 + 24 + 3) = U(36) = 6$

b)
$U( 71^{10000001} ) = 1$

c)
$U( 1009^{9001})= U( 1009^{2*4500 + 1}) = U( 1018081^{4500}*1009) = 1*9 = 9$

d)
$U( 77^{36} + 77^{37} + 77^{38} + 77^{39})=$

$= U[( 77^{36})( 77^{0} + 77^{1} + 77^{2} + 77^{3} )] =$

$= U[( 77^{4*9})( 77^{0} + 77^{1} + 77^{2}77^{3})] =$

$= U[(35153041^{9})(1 + 7 + 9 + 3)] = U[1*(20)] = U(20) = 0$