Question

Determinati ultima cifra a numerelor :
a)99 la puterea 51
b) 11 la puterea 53 +15 la puterea 53+17 la puterea 53
c) 313 la puterea 100
d) 68 la puterea 86
e)89 la puterea 37 +88 la puterea 38 +87 la puterea 39

Answer 1

a) 99^51 = 99^(4·12 +3)  Uc = Uc(9³) = 9
b) Uc = Uc(1 + 5 + 7) = Uc(13) = 3
c)  Uc = Uc(3^100) = 1
d) Uc = Uc(8^2) = 4   (86 = 4·21 + 2)
e) Uc = Uc(9 + 4 + 3) = 6

Answer 2

a. u.c[99⁵¹]=u.c[99¹]=u.c[99]⇒u.c=9;
b. u.c[11⁵³]=u.c[11³]=u.c=1;
u.c[15⁵³]=u.c[15³]u.c=5;
u.c[17⁵³]=u.c[17³]=u.c=3;
⇒u.c[1+5+3]=u.c=9;
c. u.c[313¹⁰⁰]=u.c[313⁰]=u.c=1;
d. u.c[68⁸⁶]=u.c[68⁶]=4;
e. u.c[89³⁷]=u.c[89⁷]=u.c=9;
u.c[88³⁸]=u.c[88⁸]=u.c=6;
u.c[87³⁹]=u.c[87⁹]=u.c=7;
⇒u.c[9+6+7]=u.c[22]=u.c=2;