Question

Determinati ultima cifra a numerelor: a)13 la puterea 74; b)99 laDeterminati ultima cifra a numerelor: a)13 la puterea 74;       b)99 la puterea 51;c) 11 la putyerea 53+15 la puterea 53+17 la puterea 53;   d)313 la puterea 100;e)68 la puterea 86;f)89(la put37)+88(la put 38)+37(la put 39)g)71(la put 10000001);    h)1009(la put 9001);  i)77 (la put 36)+77(la put37)+77(la put 38)+77(la put 39) ;     VA ROG CAT MAI REPEDE 

Answer 1

 
a)13 la puterea 74;
U(13⁷⁴)  = U(13² *13⁷²) = U(13² * (13⁴)¹⁸) = U(169 * 28561¹⁸) = 9 * 1 = 9
       
b)99 la puterea 51;
U(99⁵¹) = U(99 * 99⁵⁰) = U(99 * (99²)²⁵ ) = U(99 * 9801²⁵ ) = 9 * 1 = 9

c) 11 la putyerea 53+15 la puterea 53+17 la puterea 53;
U(11⁵³ + 15⁵³ + 17⁵³) = U(11⁵³ + 15⁵³ + 17 * 17⁵²)=U(11⁵³ + 15⁵³ + 17 * (17⁴)¹³)=
=U(11⁵³ + 15⁵³ + 17 * 83521¹³)=U*(1 + 5 + 7 * 1) = U(13) = 3

d)313 la puterea 100;
U(313¹⁰⁰) = U((313⁴)²⁵) = U((3⁴)²⁵) = U(81²⁵) = 1

e)68 la puterea 86;
U(68⁸⁶) = U(68² * 68⁸⁴) =   U(68² * (68⁴)²¹) = U(8² * (8⁴)²¹) = 
=U(64 * 4096²¹) = U(4 * 6) = U(24) = 4

f)89(la put37)+88(la put 38)+37(la put 39)
U(89³⁷ + 88³⁸ + 37⁸⁹) = (89 * 89³⁶ + 88² * 88³⁶ + 37 * 37⁸⁸) =
= U(89 * (89²)¹⁸ + 88² * (88⁴)⁹ + 37 * (37⁴)²²)= U(9 * (9²)¹⁸ + 8² * (8⁴)⁹ + 7 * (7⁴)²²) =
=U(9 * 81¹⁸ + 64 * 4096⁹ + 7 * 2401²²) =U(9 * 1 + 4 * 6 + 7 * 1) =
=U(9 + 24+ 7) = U(40) = 0

g)71(la put 10000001);    
U(71¹⁰⁰⁰⁰⁰⁰¹) = U(1¹⁰⁰⁰⁰⁰⁰¹) = 1

h)1009(la put 9001);  
U(1009⁹⁰⁰¹) = U(1009 * 1009⁹⁰⁰⁰) = U(1009 * (1009²)⁴⁵⁰⁰)=
U(9 * (9²)²²⁵⁰) = U(9 * 81⁴⁵⁰⁰) = 9 * 1 = 9

i)77 (la put 36)+77(la put37)+77(la put 38)+77(la put 39)
U(77³⁶ + 77³⁷ + 77³⁸ + 77³⁹) =U[77³⁶(1 + 77 + 77² + 77³)] =
=U[(77⁴)⁹ * (1 + 77 + 5929 + 456533)]= U[(7⁴)⁹ * (1 + 7 + 9 + 3)] =
= U[2401⁹ * 20] = 1* 0 = 0