Question

ultima cifra a numarului 2 la puterea 83 este...iar ultima cifra a numarului 2 la puterea 83 + 3 la puterea 84.

Answer 1

$u(2^1)=2$
$u(2^2)=4$
$u(2^3)=8$
$u(2^4)=6$
$u(2^5)=2$
Observam ca o ultima cifra a puterilor de 2 se repeta din 4 in 4 puteri
83:4=20 rest 3
Daca da rest 3 la impartirea la 4, atunci
$u(2^{83})=u(2^3)=8$

La fel se repeta si la puterile de 3, din 4 in 4
84:4=21 rest 0
Daca da rest 0, se considera rest 4
$u(3^{84})=u(3^4)=u(81)=1$
$u(2^{83}+3^{84})=8+1=9$

Answer 2

2¹=..2
2²=..4
2³=..8
2⁴=..6
2⁵=..2 => 83:4=20rest3<=>u.c.2⁸³=8
3¹=..3
3²=..9
3³=..7
3⁴=..1
3⁵=..3 => 84:4=21rest0<=>u.c.3⁸⁴=1
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u.c. 2⁸³ + 3⁸⁴ = 8 + 1 = 9