Question

Rezolvati-le Va rog ,Dau coroana

Answer 1

a)
stiind ca :
30 * 30 are ultima cif. 0
31 * 31 are ultima cif. 1
35 * 35 are ultima cif.5
=> ...5+....1+...0=6 => ultima cif.》6
b)
6^2 >> cif. 6
6^3 >> cif. 6
.... 6^40 >> cif. 6 => 40*6=240 => cif.0

c)
2^1999= 2^1998 * 2
(2^2)^999 * 2
4^999 * 2
4*998 * 4
(4^2)^499 * 4
16^499*4 =....6*4 = ....4 => 2*1999 = ....4
5^2000 =.....5
=> ....4 + ....5 + 3 = ...2 ultima cif.

d)
2^1993 =2^1992*2=16^498 *2 = ...6*2=...2
2^1993= ...2
6^1890=....6
7^1234= 49^617 (49^2n=..01 si 49^2n+/-1=...49)
=> 49^617 = ...9 => 7^1234 =...9
5^1993 = ...5
9^1564=...1
efectuam : ...2 * ...6 * ...9 *...5 *...1 =540 => ultima cif. 0

e)
3^1986= 9^993= ...9
4^1987 (4^2n=...6 si 4^2n+/-1=...4) =>
4^1987=...4
5^1988=...5
efectuam: ...9 + ...4 + ...5 = ...8 ultima cif.

2)
7^1989 = 7^1988 *7 = 49^994 *7 = ...01*7 = ...07
7^1989 =...07 ultimele doua cif.